First, you don't need to have a COVID test to visit Turkey regardless of where you come from. So, I wouldn't suggest doing a COVID test if you're not continuing to a country that requires a COVID test or your country of residence requires a COVID test upon returning.

Unless you have symptoms ( high fever, cough, etc... ) at the border control, you would not be tested at the airport without your own consent. If you don't have any symptoms, you would not be tested.

If you get tested positive, you would sent to a private hospital at your own expense, depending on bed availability, and you would be asked to self-isolate at the hospital room on your own expense as a foreigner. You would not be able to leave the hospital until your 14-day isolation has ended. Plus, there isn't a list of hospitals that foreigners are sent to.

By the way;
There is nothing related to Turkish Airlines here, so originally this thread should have gone to the Europe forum.