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We study the growth and isoperimetry of infinite clusters in slightlysupercritical Bernoulli bond percolation on transitive nonamenable graphs underthe $L^2$ boundedness condition ($p_c<p_{2\to 2}$). Surprisingly, we find thatthe volume growth of infinite clusters is always purely exponential (that is,the subexponential corrections to growth are bounded) in the regime$p_c<p<p_{2\to 2}$, even when the ambient graph has unbounded corrections toexponential growth. For $p$ slightly larger than $p_c$, we establish theprecise estimates \begin{align*} \mathbf{E}_p \left[ \# B_\mathrm{int}(v,r)\right] &\asymp \left(r \wedge \frac{1}{p-p_c} \right)^{\phantom{2}}e^{\gamma_\mathrm{int}(p) r} \\ \mathbf{E}_p \left[ \# B_\mathrm{int}(v,r) \midv \leftrightarrow \infty \right] &\asymp \left(r \wedge \frac{1}{p-p_c}\right)^2 e^{\gamma_\mathrm{int}(p) r} \end{align*} for every $v\in V$, $r \geq0$, and $p_c < p \leq p_c+\delta$, where the growth rate$\gamma_\mathrm{int}(p) = \lim \frac{1}{r} \log \mathbf{E}_p\#B(v,r)$ satisfies$\gamma_\mathrm{int}(p) \asymp p-p_c$. We also prove a percolation analogue ofthe Kesten-Stigum theorem that holds in the entire supercritical regime andstates that the quenched and annealed exponential growth rates of an infinitecluster always coincide. We apply these results together with those of thefirst paper in this series to prove that the anchored Cheeger constant of everyinfinite cluster $K$ satisfies \[ \frac{(p-p_c)^2}{\log[1/(p-p_c)]} \preceq\Phi^*(K) \preceq (p-p_c)^2 \] almost surely for every $p_c<p\leq1$.

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters