When a gas is compressed it gets hot in simple terms. The turbocharger might be hot but the amount of time the air is inside a compressor housing is not enough to heat it that substantially. Pressure law states as pressure increases, temperature increases.
Taking say 1 pound of air mass at 70F degrees and compressing it from 0psig to 14.7 psig (1 bar above atmospheric pressure), effectively raises its temperature from 70F degrees to 140F degrees simply from increasing its pressure. As was stated a turbocharger isnt cold, nor is it 100% efficient at compressing air so it would gain more heat and the temperature would be even higher. i have no real life figures so im not going to speculate the end result.
Taking that same 1 pound air mass but at 140 degrees, closer to under hood temperatures, and compressing it to the same 14.7psiG (1 bar above atmospheric pressure) effectively raises its temperature to 280F degrees.
Thats a difference of 140F degrees (again based on strictly compression only, turbocharger inefficiencies and heat gained from actual temperature of the turbocharger itself would be on top of this increase) going into your intercooler.
This is based on pressure law and real life results vary but a simple thing such as a cold air intake can help out greatly, And what i was talking about the heat generated from a turbo charger isnt a fixed amount.
Turbochargers not doing a very good job sucking is because the way a centrifugal pump works doesn't promote good suction when a restriction is set in place. A centrifugal pump uses centrifugal force to "fling" the air in between the fins out towards the housing and out of the outlet. If there is a lack of air present in the inlet of the housing the fins will have nothing to "fling" and it will just spin.
Take a vacuum cleaner and put your hand over the tube, you will or should notice the RPM of the motor increases. That is because your cutting off the air supply to the centrifugal pump and since theres nothing to pump the pump is easier to spin because its not doing any work. A centrifugal pump can only pump if there is air to displace.
In comparison to a piston pump witch directly draws in a volume of air every revolution will decrease pressure (or create a vacuum, im trying not to be too technical or too simplistic) as long as the seals stay perfectly sealed.
