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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?enter image description here

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    Turbocharges rely on very hot exhaust gases to work efficiently. Cooler exhaust gases would mean denser gases with translates to less volume of gases to spin the impeller.
    – Moab
    Jun 13, 2018 at 15:10
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    The exhaust side of a turbo can actually start to glow when driven at full power for sufficient time - seen this in a 200bhp tractor while it was working hard.... As long as it has a sufficient oil supply - quality and volume it will be fine... What are you wanting to achieve?
    – Solar Mike
    Jun 13, 2018 at 15:28

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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.

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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).

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