I have a followup to the following question: Flatness of normalization.
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in a finite separable extension of $K(X)$. Then is $X'\rightarrow X$ flat?
The motivation is that I am trying how to understand normalizations of etale covers on complements of divisors.
Thanks!