When the AC is not turned on, the radiator fan would not work till the fan switch is activated. When it is working, it would stop when the fan switch is turned off as the coolant temperature is reduced.
If the AC is turned on, the cooling fans would come on and when the compressor stops, the fan would also turn off.
Above are normal operating conoditions and there is nothing wrong with the system.
If however the fans turn off and the temperature increases, then possible causes are : 1. A failing fan motor. The carbon brushes could be wearing out and this would at tmes not allow the fan to turn on or would stop while it is supposed to work. When the fan stops, try knocking it to see if the fan restarts.
2. A failing fan relay. The contacts are not good and would intermittently fail resulting in the fan motor stopping. Try swapping the fan relays.
3. A failing fan switch that intermittently fails to contact to operate the fan relays to supply power to the fan motor. Unplugging the fan switch should tun the fan on.