All varieties here is defined over complex number. Let $X$ be a normal projective irreducible variety and let $Y$ be a projective irreducible variety. Suppose that there is a bijective morphism $f : X \to Y$. This does not imply that $Y$ is normal. One can easily construct a counterexample for curve case.

My question is:

1. Is it true if $Y$ is regular in codimension 1?

2. Is there any extra condition guarantee normality of $Y$?

I searched some related questions and answers such as

Nonsingular/Normal Schemes or

Checking whether a variety is normal ,

but I cannot find or prove a method working on my situation. In my problem, finding local defining equations is almost impossible.

Edit: One of motivation is a kind of inverse problem of Zariski's main theorem. Suppose that $g : X \to Y$ is a morphism between projective varieties with connected fiber from normal variety $X$. By taking Stein factorization, we can reduce to the situation of question.