Ok, so this is where I reveal my ignorance as an algebraic geometer: I had previously asked about pushforward of line bundles from the smooth locus of a variety to the whole thing. I think I understand basically how that picture works now, but I have a variation thereof that I would like to ask about.

Let $X$ is a normal variety (let's say irreducible quasi-projective of finite type over $\mathbb{C}$; you can even assume terminal and $\mathbb{Q}$-factorial if you like), $Y$ its smooth locus and $Z$ its singular locus. In fact, let's say the dimension of $Z$ is at most $\dim X - 4$ just for good measure.

What can be said about the local cohomology $H^i_Z(\mathcal{O}_X)$ which fills in the exact sequence $$\cdots \to H^i_Z(\mathcal{O}_X)\to H^i(X;\mathcal{O}_X)\to H^i(Y;\mathcal{O}_Y)\to \cdots?$$

What I'd love to say is that this is 0 in degrees $\leq 3$, but obviously, I'd accept other answers if they happen to be true.