No, it's not normal. I am guessing there is some wear or broken part, either the upper or the plunger, that is allowing the forward assist to rotate. Drive out the roll pin that holds it together, remove the spring, and figure out where the problem lies. If you have another forward assist to compare it to, that would probably make it even easier.
Generally speaking though, you almost never should touch your forward assist. If your bolt isn't shutting when you hit the bolt catch, then it's time for a new recoil spring. I can think of two applications of the forward assist. One, on military rifles that tend to see a lot of wear, recoil springs may not be replaced as often as they're supposed to. The forward assist is just a way of making sure that the bolt is fully seated in the carrier. The other is when you want to quietly chamber a round or chamber check, and your bolt doesn't rotate and seat into place (not all that uncommon with new guns). Otherwise for day-to-day range use, it should be of the less-used features on the AR.