It's a tough one-screen puzzle. Currently rated super expert with 1% completion rate. The following explains how to solve this level. If you'd like to try it on your own you should avoid reading. You can also stop reading after each new step so you can try starting solving on your own at that point. There are other solutions that need extra dexterity, but this is the one I initially intended.
Going to that door
There's 3 sets of doors. The door on top of the lava bridge connects to the door at the top. The two P-doors (seen as outlined rectangles) are, of course, connected to each other. There is also the door at the center that takes to the final area. The objective of the level is to be able to enter this door AND also have an active P-switch so that the P-doors can be used and you can reach the Axe.
I recommend first simplifying the level. Forget about the P-doors. Our first objective is to learn how to reach the door at the center so that we can reach that area next to the Thwomp.
Pipe object cloning
If we plan to enter that door, it appears that we need a bunch of objects below it so that Mario can stand on them and enter the door. There seems to be a notorious lack of those objects. The distance from the ground to that door is 7 squares. And counting the P switch and Pow from the two pipes, we only have 3 objects available. Something is strange here.
The first mechanic we should learn is that we can "clone" objects that come out of pipes by using doors. Hold an object, and enter the top door:
This causes you to leave holding a P-switch, but the interesting thing is the Pipe spawns another P-switch.
This trick can only clone the object you are holding while entering the door. So it works at most once. So we only have access to 4 objects.
The door and the conveyor beltThere are two main conveyor belts and they work in opposite directions. The main mechanic in this puzzle is to take advantage of this so that you can use them to build a 'bridge' between them. Like this:
So you have two objects held above above the conveyor belts and we just need one more object on top of them to enter the door. The problem is that the conveyor belts are of different lengths and speeds so it is non-trivial to put objects on them so that they meet at exactly the same time. This is where the P switch enters. Note that if you use a P switch, there's still three other objects available, enough to build our entrance. The P-switch can stop conveyor belts from moving AND also turn the two yellow blocks into coins so that we can drop objects there so that they meet.
Now all that we need to do is put a Pow block on top of a yellow block. Then press the P switch. The Pow falls on top of the conveyor belt. Be ready to drop the other P switch on the other conveyor belt.
If you time things correctly, the two objects will meet at the same time:
Now all is ready, you can get to the the center area, take that pow block and drop it below the door.
The real Puzzle
Now we just need to find a way to enter that door while a P switch is active. It's not that difficult really. If you return two steps back, just hold the Pow block and press the P switch next to the Pow block, since P-switch stops the Conveyor Belt, the other Pow will not fall, this allows you to drop the pow block on top:
Hope it was fun.