  The extremal number of a graph $H$, denoted by $\mbox{ex}(n,H)$, is the
maximum number of edges in a graph on $n$ vertices that does not contain $H$.
The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem says that for a complete
bipartite graph with parts of size $t\leq s$ the extremal number is
$\mbox{ex}(K_{s,t})=O(n^{2-1/t})$. It is also known that this bound is sharp if
$s>(t-1)!$. In this paper, we prove that if $H$ is a bipartite graph such that
all vertices in one of its parts have degree at most $t$, but $H$ contains no
copy of $K_{t,t}$, then $\mbox{ex}(n,H)=o(n^{2-1/t})$. This verifies a
conjecture of Conlon, Janzer and Lee.
