Think of it like you're in your car. If you're going 100mph, and you turn the wheel as hard as you can, you will not make a very tight turn (and probably flip your car [i.e. lots of g's]), but if you're going 20mph, you won't really be thrown to the side and will probably make a pretty tight turn. Think of the min radius turn as a 20mph turn, they're just trying to turn in as little space as possible, so they aren't going super fast and as a result don't pull a whole lot of g's.
(not a pilot, but I believe my logic is correct. Please correct me if I'm wrong)
^Thats breaking it down Barney style!