Let $X^\nu,Y^\nu$ be normalizations of affine varieties $X$ and $Y.$ If a morphism $f:X\to Y$ is a bijection, does it imply that its lift $f^\nu: X^\nu\to Y^\nu$ is an isomorphism?
We should be able to construct a counterexample as follows:
Let $Y$ be an affine curve which is smooth away from a single node. We obtain $X$ from the normalization of $Y$ by removing one of the points mapping to the node of $Y$. The map from $X$ to $Y$ is a bijection, but the map $X^\nu\to Y^\nu$ is not an isomorphism.