The first US fighter to have air conditioning was the P-80. Bleed air from the engine was run through a heat exchanger and then an air cycle machine, also called a cooling turbine, where the temperature fell, after which it was ducted into the cockpit to provide cooling. Uncooled bleed air was used to add heat to keep it from getting too cold. I believe that early system lacked a water separator, leading to snowballs being hurled into the cockpit under some circumstances..
The P-47 could have only have had outside air and air heated by the engine introduced into the cockpit. It would have been theoretically possible to tap off some of the turbo pressurized air and add a cabin outflow valve to pressurize the cockpit, but I do not think that was ever done, except possibly experimentally.