Here's another type of counterexample not (as I write) ruled out by the hypotheses of the question: consider an inclusion of fields $K\to L$, with $K$ and $L$ finite extensions of $k$. Now take the spec. Integral schemes, reduced fibres, bijective on points.
EDIT: aah, but even $k$ alg closed of char zero doesn't save you! Consider a nodal curve $Y$ and its normalisation $X'$. Now $X'\to Y$ isn't bijective, it's 2-1 at the singularity. So let $X$ denote $X'$ minus one of the points mapping to the singularity. I think $X\to Y$ is a counterexample.
Here's another type of counterexample not (as I write) ruled out by the hypotheses of the question: consider an inclusion of fields $K\to L$, with $K$ and $L$ finite extensions of $k$. Now take the spec. Integral schemes, reduced fibres, bijective on points.