Check coolent levels, pump and radiator fan. Make sure all the parts are clean and unpainted as well even the front grill.

Best way to test, in the hot sun put the aircon to full/max, how cold is it? also how hot is the radiator? If the radiator is not hot then you have a problem with the coolent/flow. A hot radiator means that the cooling system is working well since the delta between engine and radiator is lower. In modern cars the radiator is typically a centralised cooler for many different parts, so the flow bit is important too same with coolent levels, check it regularly if you are having problems as well, you can also do a coolent system flush and make sure there is no dirt or anything stuck inside.

Make sure you use both the proper coolent and right engine oil.

But what about coolent flow, make sure you check that.

Other than what one above mention where hose flatten, check that theres no blockages and check the pumps, service/change them if needed. It really just sounds like you're lacking coolent flow.

nope. Older cars the aircon is separate so requires more power from engine. Modern cars the aircon is integrated into cooling system so it takes cold from the system making it work harder.