In this paper, we study diffusions with bounded pairwise interaction. We show for the first time propagation of chaos on arbitrary time horizons in a stronger $L^2$-based distance, as opposed to the usual Wasserstein or relative entropy distances. The estimate is based on iterating inequalities derived from the BBGKY hierarchy and does not follow directly from bounds on the full $N$-particle density. This argument gives the optimal rate in $N$, showing the distance between the $j$-particle marginal density and the tensor product of the mean-field limit is $O(N^{-1})$. We use cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an arbitrary order $i$, these provide ``low-dimensional'' approximations to the $j$-particle marginal density with error $O(N^{-(i+1)})$.