  In this paper we prove that for every $s\geq 2$ and every graph $H$ the
following holds. Let $G$ be a graph with average degree $\Omega_H(s^{C|H|^2})$,
for some absolute constant $C>0$, then $G$ either contains a $K_{s,s}$ or an
induced subdivision of $H$. This is essentially tight and confirms a conjecture
of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak. A
slightly weaker form of this has been independently proved by Bourneuf,
Buci\'c, Cook, and Davies.
  We actually prove a much more general result which implies the above (with
worse dependence on $|H|$). We show that for every $ k\geq 2$ there is $C_k>0$
such that any graph $G$ with average degree $s^{C_k}$ either contains a
$K_{s,s}$ or an induced subgraph $G'\subseteq G$ without $C_4$'s and with
average degree at least $k$.
  Finally, using similar methods we can prove the following. For every $k,t\geq
2$ every graph $G$ with average degree at least $C_tk^{\Omega(t)}$ must contain
either a $K_k$, an induced $K_{t,t}$ or an induced subdivision of $K_k$. This
is again essentially tight up to the implied constants and answers in a strong
form a question of Davies.
