Let $S$ be an integral Dedekind scheme.
Let $f:X\longrightarrow \mathbf{P}^1_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme.
Let $\eta$ be the generic point of $S$. Note that $f_\eta:X_\eta\longrightarrow \mathbf{P}^1_{K(S)}$ is a finite morphism of curves over $K(S)$.
Question. Is $f$ the normalization of $\mathbf{P}^1_S$ in the function field of $X_\eta$?
In your case the function field of $X_{\eta}$ is the same as the function field of $X$. Thus the following general remark answers your question affirmatively.
Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field $K(Y)$ of $Y$. Let $\pi\colon X \to Y$ be an integral morphism of schemes such that $X$ is integral and normal and such that $\pi$ induces on function fields the extension $K(Y) \subset L = K(X)$. Then $X$ is the normalization of $Y$ in $L$. In fact this follows essentially from the definition of "normalization" and the fact that integral ring homomorphisms are stable under localization.
Yes. This follows from Zariski's Main Theorem (although there are probably more direct arguments in this case).
