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A conjecture of Alon, Krivelevich and Sudakov states that, for any graph 
$F$, there is a constant 
$c_F \gt 0$ such that if 
$G$ is an 
$F$-free graph of maximum degree 
$\Delta$, then 
$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$. 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 
$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$ 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-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.
