In this paper we present a theoretical and algorithmic analysis on the normality of rational parametrizations of algebraic plane curves over arbitrary fields of characteristic zero. If the field is algebraically closed we give an algorithm to decide whether a parametrization is proper and, if not, a normal parametrization is computed. If the field is not algebraically closed the problem is more complicated, and a degenerated situation may appear. We classify the degenerations in strong and weak degenerations, and an algorithm to decide this phenomenon is derived. Furthermore, we prove that if the parametrization is strongly degenerated then the curve cannot be normally parametrized, but weak degenerations can be resolved, and an algorithm to reparametrize the input weakly degenerated parametrization into a non-degenerated one is given. In addition, we show how these results can be applied and improved to the case of real rational curves. In this case, we present an algorithm that decides whether a given real parametrization can be normally parametrized over the reals, and that computes such a parametrization if it exists.