Its called an internal combustion engine for a reason. Spark plug sparks, sets of the fuel air mixture, explosion happens at
the top of the piston travel, and then is pushed back down by the explosion. Did I miss something in the obvious?
On the matter about the turbo needing hot air because its more active then cold air, that only matters if the air (exhaust in this cause) is not moving to begin with. The engine is a Air Pump, what comes in must come out and is forced out with the return of the piston traveling back up. That process is what forces the air(exhaust), no matter HOT or COLD, quickly through the exhaust manifold and into the turbo.
So because there was an explosion before the air could be expelled, and explosions cause heat, (that is why the gas/air/exhaust is HOT and not COLD), things get hot. This doesn't mean that Heat is needed to make a turbo run. Trying to make this as easy to understand as possible, don't get why its not.
I'm not an expert and never claimed to be, but come on, this stuff just makes sense right? We all know that HOT air is very active and violent, and will rise, And COLD air is, lets just say sleepy and likes to stay put. This has no affect on the speed at which the Turbo spools when any engine is running. If that were the case then a hair dryer would move a turbo faster then a leaf blower of equal force. They would be the same if you could control the experiment.
There, I tried to make it easy to understand the way I think about this subject
Thanks for reading.