We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., it changes the entire configuration in one step. We prove that, whenever Strong Spatial Mixing (SSM) holds, the mixing time on any $n$-vertex cube in ${\mathbb Z}^d$ is $O(\log n)$, and we prove this is tight by establishing a matching lower bound on the mixing time. The previous best known bound was $O(n)$. SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in $d=2$ dimensions throughout the high-temperature (single phase) region. Our result follows from a Modified Log-Sobolev Inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen-Wang dynamics, which extends to general bipartite graphs of bounded degree. This factorization leads to several additional results, including mixing time bounds for a number of natural local and non-local Markov chains on the joint space, as well as for the standard random-cluster dynamics.