Let $k$ be an algebraically closed field of characteristic $0$ and let $f:X \rightarrow Y$ be a map between smooth varieties over $k$ that is bijective on closed points.  Musr $f$ be an isomorphism?

The restriction to fields of characteristic $0$ is necessary to rule out the Frobenius map, and the restriction to smooth varieties is necessary  to rule out resolutions of singularities of curves.  My guess is that there will be counterexamples with $X$ and $Y$ algebraic surfaces, but I do not know a rich enough store of examples of surfaces to find one.