  We consider extremal problems for subgraphs of pseudorandom graphs. For
graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes
the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$.
Extending classical Tur\'an type results for odd cycles, we show that
$\pi_{F}(\Gamma)=1/2$ provided $F$ is an odd cycle and $\Gamma$ is a
sufficiently pseudorandom graph. In particular, for $(n,d,\lambda)$-graphs
$\Gamma$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues
in the interval $[-\lambda,\lambda]$, our result holds for odd cycles of length
$\ell$, provided \[ \lambda^{\ell-2}\ll
\frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor
this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the
condition is best possible and was proven previously by Sudakov, Szab\'o, and
Vu, who addressed the case when $F$ is a complete graph. A construction of Alon
and Kahale (based on an earlier construction of Alon for triangle-free
$(n,d,\lambda)$-graphs) shows that our assumption on $\Gamma$ is best possible
up to the polylog-factor for every odd $\ell\geq 5$.
