Clash at Demonhead/Route 11

Details
Assuming you first enter this Route from the left exit, shoot out the enemies which float up and down above the water, and then jump in. You'll need to swim all the way to the right, ignoring more enemies which appear above you. Once you reach the other side, watch out for a few small enemies that come running toward you, followed by one more floating enemy before you get to climb up to the screen above.

You'll encounter a set of large humanoid enemies carrying guns. While they can shoot you, they are very large and predictable targets, so they are pretty easy to eliminate. Clear the short path to the left, and then jump up through the platform and continue on to the right. Shoot out the rest of the humanoids that you encounter, and look for a gap in the ceiling of the cavern above you. Jump up through the gap to each the next screen.

When you arrive, you'll see four heat-seeking missiles headed toward you. Don't move, and all four will collide with the ledge above you and explode harmlessly. Then hop up and around and continue to the right. Watch out for a few more missiles as you approach the door to the governor. Fuel up on Ultra-food to restore any lost health, and summon a shop in case you need to buy more.

Governor: Gajara
When you enter the room, you must wait a brief moment before Gajara speaks to you. When he does, he says he's been waiting for you. You ask about Professor Plum, and Gajara explains that he is in a hypnotic state, working on a bomb. Then he attacks you.

You can activate a Guard Crystal, or use Big Shot, but the truth is, you don't really need to use either. Gajara's behavior is very predictable, and you can use that against him. Shoot two bullets at him as soon as you can. When the first bullet strikes his head, he stops advancing, and begins firing a series of fireballs as he leaps into the air. The second bullet should take out the lowest fireball that he spits out. The next fireball above it can be ducked under.

He will come down spitting more fireballs. As he lands, you want to fire two shots; one to strike him in the head and cause him to rise again, and the second to destroy his first fireball. Squat back down to duck beneath two low fireballs, one from the end of his last trip down, and the other from the start of his next trip up. Repeat this process several times until you hit him 15 times and he is defeated.

If Gajara gets too close to you, and you don't have enough time to shoot and duck, consider running beneath him as he leaps into the air, and shooting him as soon as he turns around after hitting the wall. Once he is defeated, you will obtain a medallion, and all of your health will be restored.