Ideally radiator fan runs automatically to bring the coolant temperature within acceptable limit. if its running for longer period, also check your engine oil level in the car.
usually when you are using the car AC the fan runs more frequently and keeps the coolant temperature in acceptable limits.
I have observed that older, lower grade or used engine oils make the car engine run bit hotter since they dont cool the engine optimally.
first step should be to check the engine oil level
2nd step change engine oil
3rd step check thermostat valve if its working properly
your automatic cooling system seems to be working fine considering that its doing the job of bringing the coolant temperature within acceptable operating range