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

https://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689 or

https://mathoverflow.net/questions/60097/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.