Suppose $X \rightarrow Y$ is a map of reduced connected projective schemes of finite type over an algebraically closed field of characteristic 0, where $Y$ is a smooth connected curve. Let $Z \rightarrow X$ be the normalization of $X$. Then, for a general closed point $y \in Y$ (meaning for all but finitely many closed points of $Y$), is $Z_y$ (the fiber of $Z$ over $y$) the normalization of $X_y$?
Edit: This question was originally asked without the characteristic 0 hypothesis. Jason Starr pointed out in the comments that it fails in characteristic $p >0$, essentially because generic smoothness does not hold, as is witnessed in the case of quasi-elliptic fibrations.
Feel free to ignore the following, but in case it helps provide context, I will now explain why I want to know this is true for my research. For other reasons, I am trying to prove the following statement:
Suppose we have a proper flat map of quasi-projective reduced schemes over an algebraically closed field $X \rightarrow Y$ where $Y$ is a smooth connected curve. Assume further that the fiber over every point in $Y$ has two irreducible components, and the fiber over a particular closed point $y \in Y$ has two irreducible components with distinct Hilbert polynomials. Then, $X$ has two irreducible components.
Essentially by considering the normalization of $X$, and using Stein factorization, I have reduced the problem to the question above (showing that the normalization of the fiber is the fiber of the normalization for a general closed point of $Y$). However, I'm stuck on this detail.