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Let $N$ be a nef Cartier divisor and let the coherent sheaf $\mathcal{F}$ have support dimension $d$. Then $h^i(X,\mathcal{F}(mN))=O(m^{d-i})$ for $1\le i\le d$. This does not require reducedness or smoothness.

Choose a very ample $A$ and fix $a$ sufficiently large for Fujita vanishing. Choose a section of $aA$ avoiding the associated points of $\mathcal{F}$. Its multiplication gives $0\to \mathcal{F}\to \mathcal{F}(aA)\to \mathcal{Q}\to 0$, where $\mathcal{Q}$ has support dimension at most $d-1$. After tensoring with ${\mathcal{O}}_X(mN)$, the middle term has zero higher sheaf cohomology, uniformly in $m$. Hence $h^i(\mathcal{F}(mN))\le h^{i-1}(\mathcal{Q}(mN))$. Induct on support dimension: for $i=1$ use the polynomial bound for sections of a fixed divisor, and for $i>1$ use the induction hypothesis. Degree greater than $d$ vanishes by Grothendieck vanishing.

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.

First suppose $X$ is integral. Induct on its dimension of a scheme. The previous two parts handle all $j>1$ and also handle $j=1$ when $h^0(X,mD)$ eventually vanishes. Otherwise choose $r>0$ with a nonzero global section of $rD$. On an integral variety it defines an effective Cartier divisor $E$, possibly empty, and multiplication by the section givesThus $h^1(X,mD)\le h^1(X,(m-r)D)+h^1(E,mD)$. For $n\ge 2$, the induction hypothesis bounds the last term by $O(m^{n-2})$; summing on each residue class modulo $r$ gives $O(m^{n-1})$. For $n=1$, $E$ is zero-dimensional and its positive-degree sheaf cohomology vanishes, so the same recurrence is bounded.

For a general projective scheme, a nonzero section can be a zero divisor, so that argument requires an additional step. The general cohomology growth for nef twists supplies it: for every coherent sheaf $\mathcal{F}$ with support dimension $d$,Its proof uses Fujita vanishing, an ample section avoiding the associated points of $\mathcal{F}$, and induction on support dimension. Taking $\mathcal{F}={\mathcal{O}}_X$ gives the required $O(m^{n-1})$ estimate for every $j\ge 1$, including nonreduced and reducible schemes; degrees above $n$ vanish.