Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.

Can I say something about the fibers over non-closed points?

Is it true that $X$ is normal?

In the local setting, a positive answer to 2 can be found in Matsumura "Commutative ring theory", Theorem 23.9 and its corollary. However, the normality of all the fibers is required.

I am interested, in particular, in the case where the fibers over the closed point are all isomorphic to $\mathbb{P}^1_{\mathbb{C}}$.