Intercooler after the engine and before the turbo

Question

what would hapen if right after the engine the exaust gas went to an intercooler and then to the turbine???

the turbine would run cooler but it would be better?

Answer 1

Cooling any gas down increases its density and, therefore, inside a closed pipe, its pressure decreases. Less pressure means less gas velocity. Less velocity, in turn, means less boost. With your approach, you would be wasting the exhaust's kinetic energy instead of using it (which is what the turbocharger was meant for, originally). Therefore, if you want the turbo to work cooler, you have to cool it directly.

Answer 2

You'd also run into the issue of the turbo producing turbulent gasses, which are more excited, atomically, than cooled gasses, and therefore carry more heat energy and less density (as stated above).