  A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph
$F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of
maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon,
Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$
that includes all bipartite graphs. Moreover, it follows from recent work by
Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $\chi(G)
\leq (t + o(1)) \Delta / \log\Delta$ as $\Delta \to \infty$. We improve this
bound to $(1+o(1)) \Delta/\log \Delta$, making the constant factor independent
of $t$. We further extend our result to the DP-coloring setting (also known as
correspondence coloring), introduced by Dvo\v{r}\'ak and Postle.
