  One of the cornerstones of extremal graph theory is a result of F\"uredi,
later reproved and given due prominence by Alon, Krivelevich and Sudakov,
saying that if $H$ is a bipartite graph with maximum degree $r$ on one side,
then there is a constant $C$ such that every graph with $n$ vertices and $C
n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the
constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in
terms of $r$. We conjecture that this is essentially the only situation in
which F\"uredi's result can be tight and prove this conjecture for $r = 2$.
More precisely, we show that if $H$ is a $C_4$-free bipartite graph with
maximum degree $2$ on one side, then there are positive constants $C$ and
$\delta$ such that every graph with $n$ vertices and $C n^{3/2 - \delta}$ edges
contains a copy of $H$. This answers a question of Erd\H{o}s from 1988. The
proof relies on a novel variant of the dependent random choice technique which
may be of independent interest.
