  Let $\Pi$ be a projective plane of order $n$ and $\Gamma_{\Pi}$ be its Levi
graph (the point-line incidence graph). For fixed $k \geq 3$, let
$c_{2k}(\Gamma_{\Pi})$ denote the number of $2k$-cycles in $\Gamma_{\Pi}$. In
this paper we show that
  $$
  c_{2k}(\Gamma_{\Pi}) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n
\rightarrow \infty.
  $$
  We also state a conjecture regarding the third and fourth largest terms in
the asymptotic of the number of $2k$-cycles in $\Gamma_{\Pi}$. This result was
also obtained independently by Voropaev in 2012.
  Let $\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$ denote the
greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and
girth at least 6. As a corollary of the result above, we obtain
  $$
  \text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) =
\left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty.
  $$
