The Great Giana Sisters/Stages 29-33

Stage 32
When you see the diamond and the pit below, jump down. The pit has a subtle blue line, normally reserved for bonus levels, but is actually the safe route.

If you instead continue right, you have to slip past the two jumping fish (possible but impractical without powerups), go through the corridor to the right (the upper two have monsters, and require a powerup to pass), and run across the rope bridge to the right. When you reach the right edge, you will discover that there is no exit, and the only option is to die and try the level again.

Stage 33
When you start, head directly right, and jump over the dragon. It will be instantly killed, allowing you to grab the gem.

If you hesitated, the dragon will be slightly more aggressive. However, you can still jump on the two blocks at the left, and make a high jump over that dragon.