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Choose very ample divisors $H_2$ and $H_1\sim D+H_2$, with defining sections avoiding the associated points of $X$. Such a choice is possible after taking a sufficiently high ample twist. Put $R_m=mD-H_1=(m-1)D-H_2$. The two indicated short exact sequences of sheaves and their long exact sequences in sheaf cohomology implyWe abbreviate $h^i(Z,{\mathcal{O}}_Z(L))$ by $h^i(Z,L)$. Each $H_a$ has dimension $n-1$ and the restricted divisor is nef. For positive degrees below $n-1$, the assumed induction estimate bounds the error terms by $O(m^{n-2})$. The displayed hypothesis omits top-degree cohomology; use top cohomology boundedness for nef twists for that degree, and Grothendieck vanishing above it. These give the same error bound. The auxiliary top-degree result follows from Fujita vanishing by cutting a coherent sheaf with an ample divisor, as proved in cohomology growth for nef twists.

For $n\ge 2$, summing the inequality over successive $m$ gives $h^j(X,mD)=O(m^{n-1})$ for every $j>1$. For $n\le 1$, all such groups vanish by Grothendieck vanishing.