My car, the air conditioner turns off when the heat fan is turned on,

Seems normal , what are you going to do run the AC and the heat at the same time ?

I misunderstood, the fan of the car is related to the engine temperature, not the heating fan of the air conditioner, the air conditioner cooling does not work when the engine fan starts to work.

Hi beatupchevy. Running the air conditioning system with the heat actually does have a purpose. It prevents the windows from fogging up in cool damp weather. I do it all the time. That's why the air conditioner compressor comes on when you select defrost. On many vehicles with automatic climate control the air conditioning compressor runs anytime the outside temp is above about 35°F unless you turn it off. It also helps keep the seals in the system from drying out and leaking from non use. Jim