This is more to the original poster and an alternative to the walking analogy. Next time you are driving in an empty parking lot, turn the wheel a ¼ turn and drive at 5 mph. You will be driving a tight circle and you will complete that circle quickly. Then step on the gas but keep the wheel where it is. As you pick up speed, your forward velocity (speed) will increase while your horizontal (wheel angle) remains constant. This causes your car to drive in larger circles than when you were only moving at 5 mph. That explains the radius/speed relationship.

The rate/speed relationship occurs because you have to travel a longer distance to travel at a higher speed because your radius is large. Think about a race on an oval track. Those race cars can take up to 15 seconds to turn 180*, but I can easily turn 180* in less than 5 seconds in my 10 year old civic in a parking lot. That is all because I am traveling slower which generates a smaller radius which creates a shorter circumference that I must travel. That is not a perfect analogy because steering angles (bank) are not the same in those too cases, but when you do it in a parking lot, you will notice that it takes longer to do a circle fast than slow with the same steering angle.

Once you get that in your head, then you can have some fun trying to figure out if it is better to get out of a box canyon by turning at Va pulling maximum Gs, or pulling flaps and turning with the stall horn going off. Of course, if you want to think outside the box to get out of the box, do a half spin and recover .