My jag s type r is over heating and the fan is staying on for a long period of time before shutting off

Could be one of the following:
Physically look in the radiator under the cap for coolant level, and check the valve in the cap for sticking. If air is present in the radiator it can't get out, and more coolant can't get in,(from the overflow). Either clean the cap or replace it and check that it is the CORRECT cap.
Radiator fins dirty, clean with a strong stream of water, not high pressure water.
Radiator clogged, have it boiled out, or replace.
Thermostat stuck open or shut, replace it.
Water pump worn out, can no longer move enough coolant, replace it.
Fan shroud broken or missing....
Electric Fan(s) not working, Check the fan, relay, fuse and engine temperature sensor's.
Belt driven fan, belt slipping, fan clutch is bad, fan blades have flattened out.

Also, if the Jag has A/C and electric fans, one fan is dedicated to the A/C and should come on almost the instant the A/C is turned on. The engine fan will run even with the key off, that is normal. The system is trying to cool itself. Hope this helps.