Regular Tur\'an numbers of complete bipartite graphs

Let $\mathrm{r}\mathrm{e}\mathrm{x}(n,F)$ 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 $\mathrm{r}\mathrm{e}\mathrm{x}(n,F)$, 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 $t>s!$.