Get your electrical wiring checked related to fan and its switch. Park the car outside. Start the engine and AC and open the bonnet. Observe the radiator Fan closely. If it is working keeping looking at it until the temperature needle reaches Mid section. See if the fan still working or stopped or even slowed down when the needle further rises to Hot mark?
2ndly check your radiator for rust inside thats blocking the columns. Get your radiator washed out or better install new radiator with dual columns. Usually washing out old radiators start new leaks and cause much headache.
3rdly, check coolant/water levels are they reducing in the reservoir plastic bottle? if yes then its about the leak in cooling system
4th, check your engine thermostat valve for correct functioning.
First 2 points mentioned above could hold the cause of your issue at hand most likely.