Suppose $X$ is a variety with mild singularities (say terminal singularities), and $W$ is a center of some exceptional divisor over $X$ (i.e. there is a variety $Y$ with birational morphism $f: Y \to X$, and a divisor $E \subseteq Y$, such that $f(E) = W$). Suppose $W$ is normal (in my mind, $W$ is a minimal log canonical center).

Question: Is there a resolution $g: \tilde X \to X$ which extracts the divisor $E$ and a subset $\tilde W \subset E$, such that the natural morphism $g|_{\tilde W}: \tilde W \to W$ is birational?

If the answer is no in general, is there any special case this holds? (I cannot assume that $X$ is smooth where the claim is certainly true.)