  Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex
graph that is regular and does not contain $F$ as a subgraph. We give lower
bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor,
when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
