my engine runs hot with air conditioner on

Cooling fans come on when A/C is turned on?

The condenser is sitting in front of the radiator. The condenser is trying to remove heat from the compressed hot refrigerant gas to condense it down to a liquid. That heat is transferred by air from the condenser through the radiator. I'm sure the radiator is just soaking some of that heat in. It shouldn't make it get close to overheating though, but I see how it's possible to raise the temp of the coolant. If it is close to overheating, check the operation of the cooling fan(s).