  Motivated by a longstanding conjecture of Thomassen, we study how large the
average degree of a graph needs to be to imply that it contains a $C_4$-free
subgraph with average degree at least $t$. K\"uhn and Osthus showed that an
average degree bound which is double exponential in t is sufficient. We give a
short proof of this bound, before reducing it to a single exponential. That is,
we show that any graph $G$ with average degree at least $2^{ct^2\log t}$ (for
some constant $c>0$) contains a $C_4$-free subgraph with average degree at
least $t$. Finally, we give a construction which improves the lower bound for
this problem, showing that this initial average degree must be at least
$t^{3-o(1)}$.
