Two basic things, the main issue is general cooling system condition and operation. You might take it to a radiator shop and ask them to pressure test it. They usually charge nothing for this. When the a/c is on it is in hot weather and the condensor in front of the radiator adds heat to the air that makes its way to the radiator, so it is naturally going to run a little warmer than in the non a/c months.

Secondly with the air conditioning turned on and stabilized, the aux fan should come on.

How hot is it getting?

Good luck,