How much of the landing runway is mine?
Everything in Dave's answer is entirely correct.  To add to it, AIM 4-3-20 says:


  The following procedures must be followed after landing and reaching taxi speed. Exit the runway without delay at the first available taxiway or on a taxiway as instructed by ATC.


(emphasis mine)

Controllers are not supposed to give pilots a turn off the runway until they've already reached taxi speed, but at busy airports, I've seen that rule stretched. You don't have to comply with any ATC instruction that you deem unsafe, but if you do and you slide off into the grass, it's your fault.