ag.algebraic geometry - Criteria for a map of schemes to be an isomorphism

Let $ X, Y $ be separated finite type schemes over an algebraically closed field $ k $. Assume that $ Y $ is reduced. Let $ phi : X rightarrow Y $ be a morphism of schemes. Suppose that $ phi $ gives a bijection on $ k $-points and an injection on $ S$-points for all $k$-schemes $ S$. Prove or disprove that $ phi $ is an isomorphism (or add some extra hypotheses to ensure that $ phi $ is an isomorphism).

When $ k = mathbb{C} $, $ X $ is normal, an $ Y $ is normal and irreducible, then I have a proof which uses the following crazy fact:
If $ X $ and $ Y $ are irreducible varieties over $mathbb{C} $ and $ Y $ is normal, then a morphism $ phi $ inducing a bijection on $ mathbb{C} $-points is an isomorphism.
My original question follows from this fact via a small tweaking of the usual Yoneda argument.

If anyone can give me a proof or reference for this last fact, I would be grateful too. I read it in the appendix of Kumar's book on Kac-Moody groups.

Edit: In light of some counterexamples, let me assume that X is irreducible and Y is normal and irreducible.