In positive characteristic, you get a counterexample by taking $X=Y=Z=$ the affine line (say), $f$ the identity and $g$ the Frobenius map.

Assume now that the ground field is algebraically closed of characteristic zero. Consider the map $(f,g):X\to Y\times Z$. Its image $\Gamma$ is a closed subvariety of $Y\times Z$. The assumption on the fibers exactly means that both projections from $\Gamma$ to $Y$ and $Z$ are bijective (on closed points). Since they are proper they must be (in char. zero) finite birational, hence (by normality) isomorphisms. So, $\Gamma$ is the graph of the isomorphism we are looking for.