It's possible that your cooling fan is NOT activating when the AC is turned on---there are instances where it should activate---see below for details:
Vehicles Equipped with AC: In addition to using coolant temperature and
battery temperature sensor to control cooling fan operation, the
cooling fan will also be engaged when the ,air conditioning system is
activated. The relay is also energized when air conditioning is
selected and coolant temperature is above 95°C (203°F) , or, air conditioning is selected and battery temperature sensor is above 41°C (106°F) . It will then de-energize when air conditioning is selected and coolant temperature is below 92°C (198°F) , or air conditioning is selected and battery temperature is below 38°C (100°F).