  The problem of Zarankiewicz asks for the maximum number of edges in a
bipartite graph on $n$ vertices which does not contain the complete bipartite
graph $K_{k,k}$ as a subgraph. A classical theorem due to K\H{o}v\'ari, S\'os,
and Tur\'an says that this number of edges is $O\left(n^{2 - 1/k}\right)$. An
important variant of this problem is the analogous question in bipartite graphs
with VC-dimension at most $d$, where $d$ is a fixed integer such that $k \geq d
\geq 2$. A remarkable result of Fox, Pach, Sheffer, Suk, and Zahl [J. Eur.
Math. Soc. (JEMS), no. 19, 1785-1810] with multiple applications in incidence
geometry shows that, under this additional hypothesis, the number of edges in a
bipartite graph on $n$ vertices and with no copy of $K_{k,k}$ as a subgraph
must be $O\left(n^{2 - 1/d}\right)$. This theorem is sharp when $k=d=2$,
because by design any $K_{2,2}$-free graph automatically has VC-dimension at
most $2$, and there are well-known examples of such graphs with
$\Omega\left(n^{3/2}\right)$ edges. However, it turns out this phenomenon no
longer carries through for any larger $d$.
  We show the following improved result: the maximum number of edges in
bipartite graphs with no copies of $K_{k,k}$ and VC-dimension at most $d$ is
$o(n^{2-1/d})$, for every $k \geq d \geq 3$.
