4-1 The Pit
Introducing us to this new, fourth world is the usual "Pit" level. Moving around this level you probably notice something strange, as you move the time for all other objects flow with you. Moving right moves time forward, moving left moves it backward.
Move as far right as you can to make the monstar grab the key. Notice how the green-glowing items are still unaffected by time manipulation. Jump down and steal the key from the unsuspecting monstar, rewind time the usual way and move through the gate. As there are no puzzle pieces on this level, make your way through the exit door.
4-2 Jumpman
The start of this level will surely feel familiar to any fans of the old Donkey Kong. Do like Mario would've done and begin to climb your way up toward the big monkey at the top. When you've climbed up two ladders, you might notice that the third will be blocked off by a monstar as soon as you try to climb it. You will need to find another way around, so go back to the second ladder and jump up the small platform beside it. Now, jump to your left and land on the monstar falling down this side. You should be able to bounce off him several times all the way up to the monkey cannon as long as you move ever so slightly left while bouncing. Up here it's simply a matter of grabbing 
Puzzle Piece 4-2-1.
From here you can probably see two locked gates with a puzzle piece between, so let's go find a key for them. Jump down to the ground level again and continue right. Move up the first ladder you get to and make your way up to the key, jumping over any fireballs. Remember, the fireballs only move when you do, so take your time going up. When you've reached the key at the top, grabbing 
Puzzle Piece 4-2-3 is a piece of cake, just jump down next to the cannon and take it.
With the key in hand you can now rewind time to the point where you took the first puzzle piece. Do not try to use it on the right hand side gate. If you try to open that gate, the gate will open, but as you'll be moving left, time will rewind and it will immediately close again. The key will be broken and used up, since it was green-glowing and not affected by time manipulation. So use it on the left gate and take the final piece of the level: 
Puzzle Piece 4-2-2. Now, you can move out and continue your journey right. A claw will stop your path if you try to reach the exit underneath the fireball obstacle course, so climb into it and jump over a cannon to reach the other side of the claw as well as the exit.
4-3 Just Out of Reach
Start to climb up the first ladder you see when you enter the level. Climbing all the way up it is tricky though, because a monstar will block the path for you. Instead, jump of at the platform with the lever on it. Flip it and nothing will happen; you will have to make it work by moving right. As you move right you might notice that the monstar will move past the ladder this time, so you're free to climb up it. But this time, the very same monstar is blocking the path to the puzzle piece. If only you could un-flip that lever somehow... Move on to the left, jump down on the falling monstar and use it to bounce back up to the same platform you jumped from. You should now have guided Tim so far left (backward in time) when you jumped that the switch will have un-flipped itself. Now you can use the monstar which was blocking your path to jump over the barrier sealing you off from 
Puzzle Piece 4-3-1.
When you've reached the previous puzzle piece, you might wonder where the next one is. See that ladder on Tim's right leading up into the sky? It's up there. Unfortunately it's sealed off by a gate, so we will have to get the key located in the crawlspace under the sky-reaching ladder. Tim is too big to reach it, only the monstar falling down close to you can pick it up from there. If you try to jump down you'll notice that either you kill the monstar, or the monstar kills you in the process. We need the monstar alive, and the trick to keeping him alive is to kill him as far to the right in your jump as you can. When he dies you'll most likely be bounced up to the platform with the ladder on it. Carefully move to the left edge of that platform (Tim can change his direction mid-air, so maybe even feign-jump of its side, and move on back.) When you move left enough, the monstar will be brought back to life. Move right and let him get the key for you. On the other side of the crawlspace, jump on him and get the key. Go back and climb the ladder to the sky, open the gate and collect 
Puzzle Piece 4-3-2.
4-4 Hunt!
It's another "Hunt!" level! The mechanic to get the puzzle piece is just as in the previous one; kill all the monstars and the gate will open. However, if you try to kill all the same way as you did before—just picking them off as you make your way up the structure—you'll notice they come back to life as you start going left. Time is a cruel mistress.
The solution is to take note of the position of the monstars when Tim is at his leftmost position on the map, for instance when at the entry door. When Tim is at this position, the monstar which is furthest to the left is the one you have to deal with firstly. Then deal with them in order going right, the furthest right one being the last.
The leftmost one is the monstar at the very top of the level, so climb all the way up and jump on him. The next two shared almost the exact same position on the map when Tim was at his leftest, so which one to kill first might be a bit tricky to figure out. You will have to check which one will be furthest lift when Tim would kill it. As it turns out, that is the one on the platform just below the first one you killed. Climb down and jump over to him. You can use the monstar on the platform below to clear the big gap between Tim and him, just like when going up to the top. The monstar you bounce on won't die since you're moving left. Then dispose of the targeted monstar.
Next is the third one. You can't go down the ladder since you will go too far left and resurrect the monstars already killed. Go right, drop down to the level below and jump over the gap and kill your third monstar. On the way back right you can drop down onto the lowest-most monstar and dispose of him too.
There's only two monstars left, and getting rid of the fifth one is as easy as walking over and jumping on it. But how can Tim make up to the last one's platform then? As long as you're moving left, the monstars won't die, so if you continue moving slightly left when jumping on a monstar, you can use it to gain significant jump boosts. Get on the right hand side of the lower monstar and jump on his head—bouncing to the left—then try to guide Tim right mid-air and land on him again. This should give enough of a boost to be able to jump up to the last surviving monstar. Jump on him too and the gate will open. The gate is green-glowing, so you can now move left again and go pick up 
Puzzle Piece 4-4-1.
4-5 Movement by Degrees
Three claws greet you at this level. All of them are glowing green and as such move freely in and out their pipes, regardless of Tim's movement. To the left of you is a key which will be used to open the gate at the top of screen. Tim is too big to get the key himself, so you will have to guide the monstar through the path of claws so he can grab it for you.
The two first claws the monstar need to pass by are simply a question of timing right. While Tim moves right the monstar will move left. So when both claws are down, make a run for it, and remember to jump over the monstar as he crosses your path. Don't worry if he gets killed by the third claw; you can just walk back a bit and he'll come back to the living. Climb up the ladder and jump off on the floating bit of pillar on Tim's left. Now wait for the right moment for the monstar to pass the last claw. When the claw is down, jump on over to the right platform and continue on. When the monstar reaches the key, you'll hear a "ding" sound. Go back to the area where the claws were located and pick up the key. Climb up to the gate, open it and then grab 
Puzzle Piece 4-5-1.
The second piece is a fair bit trickier to get at. The roles will be reversed for the enemies this time; Tim can control the movement of the three claws, but the monstars are unaffected by time manipulation. Go back to the place where you heard the ding from the key. You can see monstars popping out into a field of claws. By moving left and right you can control whether the claws are up or down, and thus guide a monstar past them. However, you can't move up the ladder fast enough to get them past the third claw. What you need to do is plan ahead for the third one. Climb up the ladder and get of it on the left. Move to a position on the platform above the square monolith where the third claw is all the way down its pipe and wait for a bit. Now jump down and guide a monstar through the first two claws. As soon as it passes the third one, very quickly rewind time to the point where you were waiting with the third claw down. The monstar should be able to pass it if you were quick enough. Then just wait for it to fall down to Tim's level, jump on it and bounce up to 
Puzzle Piece 4-5-2. That was all of them for this level so carry on to the next.
4-6 Movement, Amplified
4-6 Movement, Amplified is—as the name implies—basically a souped-up version of 
4-5 Movement by Degrees. For the first puzzle piece you will have to grab the key on Tim's left, but as in the previous level, you need to use the monstar to get it. Carry on right and the monstar will move left. Climb over the first two claws you encounter and guide the monstar past the other two. Now, getting the monstar through the path of the other two claws is nigh impossible; they're just too fast. What you have to do is climb up the tall ladder to the lower cloud bridge above the claws. As the cloud bridge is moving in the same direction as Tim is, the speed in which Tim can move—and thus the speed time can move—will be amplified. While moving on the cloud bridge you should be fast enough to guide the monstar through the second clique of claws as well. Continue right until you hear a "ding" sound and go back and pick up the key. Climb up to the top cloud bridge and use the key on the left gate (it will break on the right hand side gate) and get 
Puzzle Piece 4-6-2.
The second piece is not too hard to get your mittens on. Make your way to the right side of the right hand side gate. From there run to your right and make a leap of faith off the very edge of the cannon while continuing going right. If your timing for the jump was right, you should land on a fast-moving cloud suddenly popping into frame. Run along it and jump off its edge to the floating platforms and maneuver to 
Puzzle Piece 4-6-1. There is a puzzle canvas beneath you which you can play around with if you want, but as you lack the last two puzzle pieces, you may continue through the door to the next level.
4-7 Fickle Companion
This is one tricky level. When you enter the level you will spot a key in front of you. This key you will have to use to open up three different gates. With time manipulation that wouldn't be such a large problem, because as you can see, the key isn't glowing green. Go pick up the key and then try to go back to the door you came in through with it. You can't! Tim can only keep a grasp on the key while going forward in time; that means going to the right. A tricky predicament indeed, and it requires a bit of planning, so you might want to familiarize yourself with the level first, before you pick up the key and wander around with it.
The easiest puzzle piece to pick up first is actually the one furthest away. Pick up the key and jump over the fire pit and climb up the first ladder and go over to the lever, but don't flip it (this will just come in handy when trying to grab the piece above later). Jump down to the monstar coming up and kill him. Quickly, grab the key if you dropped it and run and open the first gate. Go past it and drop down to the crawlspace underneath the gate and move as far left as you can in there. If you were quick enough, the gate should have closed, the key become whole and a new monstar should have grabbed hold of it. Wait until the monstar goes into the area you are now locked into, steal his key and use it to open the gate to 
Puzzle Piece 4-7-1.
Now you can rewind time to the point when you were next to the lever, after jumping over the fire pit and climbing the subsequent ladder. Flip the lever and walk right, dropping down to the left of the crawl space where the monstar roam. A platform activated by the lever will start to move with you. Kill off the monstar down here and a new one will spawn from the cannon up top. Try to catch the new monstar mid-air on the platform. As soon as he lands on it, quickly move left and escort him to the platform with the lever on it. Rush before him and climb up the ladder to the same platform as him. Wait until he gets past the ladder leading up to the gate and steal the key he picked up. It should now be clear sailing going up and grabbing 
Puzzle Piece 4-7-2, the last puzzle piece in this world. Go back to the first gate you opened and climb the tall ladder next to it, then go right to find this world's castle.
The greeter at the end of this level is affected by time being dependant on Tim's position, and so will stop before saying that the princess is in another castle, and will say "Hey! Where are you going?"