IMO, one important thing to remember to answer this question is that a turbo changes 'velocity energy' into 'pressure energy' which creates heat. The turbo as I understand it, has only one job.
Stands to reason, cooler denser air is the job of 'a' cooling system (not the turbo) whether its intercooler method, intracooler method, water injection, or whatever. Believe me, everything has been tried to cool Turbo air pressure heat generation. A good book in the HPBOoks Turbocharges by Hugh Macinnes.
Ferd