  Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$
contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of
$G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove
that $c=\frac{3}{8}$. More generally, we show that for any $\varepsilon>0$, and
any integer $k \ge 2$, there is a $C_{2k}$-free graph $G_1$ which does not
contain a bipartite subgraph of girth greater than $2k$ with more than
$\left(1-\frac{1}{2^{2k-2}}\right)\frac{2}{2k-1}(1+\varepsilon)$ fraction of
the edges of $G_1$. There also exists a $C_{2k}$-free graph $G_2$ which does
not contain a bipartite and $C_4$-free subgraph with more than
$\left(1-\frac{1}{2^{k-1}}\right)\frac{1}{k-1}(1+\varepsilon)$ fraction of the
edges of $G_2$.
  One of our proofs uses the following statement, which we prove using
probabilistic ideas, generalizing a theorem of Erd\H{o}s: For any
$\varepsilon>0$, and any integers $a$, $b$, $k \ge 2$, there exists an
$a$-uniform hypergraph $H$ of girth greater than $k$ which does not contain any
$b$-colorable subhypergraph with more than
$\left(1-\frac{1}{b^{a-1}}\right)\left(1+\varepsilon\right)$ fraction of the
hyperedges of $H$. We also prove further generalizations of this theorem.
  In addition, we give a new and very short proof of a result of K\"uhn and
Osthus, which states that every bipartite $C_{2k}$-free graph $G$ contains a
$C_{4}$-free subgraph with at least $1/(k-1)$ fraction of the edges of $G$. We
also answer a question of K\"uhn and Osthus about $C_{2k}$-free graphs obtained
by pasting together $C_{2l}$'s (with $k>l\ge3$).
