  We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski,
Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$
such that for every positive integer $s$, every graph of average degree at
least $p(s)$ contains either $K_{s,s}$ as a subgraph or contains an induced
subdivision of $H$. This improves upon a result of K\"uhn and Osthus from 2004
who proved it for graphs whose average degree is at least triply exponential in
$s$ and a recent result of Du, Gir\~{a}o, Hunter, McCarty and Scott for graphs
with average degree at least singly exponential in $s$. As an application, we
prove that the class of graphs that do not contain an induced subdivision of
$K_{s,t}$ is polynomially $\chi$-bounded. In the case of $K_{2,3}$, this is the
class of theta-free graphs, and answers a question of Davies. Along the way, we
also answer a recent question of McCarty, by showing that if $\mathcal{G}$ is a
hereditary class of graphs for which there is a polynomial $p$ such that every
bipartite $K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most
$p(s)$, then more generally, there is a polynomial $p'$ such that every
$K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most $p'(s)$. Our
main new tool is an induced variant of the K\H{o}v\'ari-S\'os-Tur\'an theorem,
which we find to be of independent interest.

Combinatorics