On the turbocharger videos, exhaust gas heat is used to pressurize the impeller then the diffuser converts velocity to pressure on the compressor side; while the turbine runs off the heat it expands to turbine while pressure of the exhaust does very little. A diffuser is a restriction. That’s why the turbocharger is a snail shaped instead of a cylinder shape.
Thanks for your time.
Matrix3281
should go into the offtopic section 
No, it’s fine where it is, he asked me before posting where to put it. 
It’s probably just the fact that it isn’t framed as a question, which is a bit confusing - along with the formulation.
The premise of the OP is to point out a mistake in the turbocharger tutorial videos. I still don’t get what is supposed to be wrong though.
Heat cannot pressure anything by itself, and we never claim it does (I just rewatched the video). The high-energy exhaust gases are hot and dense, which means that they exhibit a strong pressure on their surroundings and (on average) move towards/expand strongly towards less hot and less dense surroundings. The turbine is the gatekeeper taking some of that energy out of the system to put it into the compressor. That is what is said in the video, paraphrased. So what exactly is wrong with that? We know that this description isn’t very detailed, but we believe it to be correct in principle.
Cheers!