  Let $\Theta_{k_1,\cdots,k_\ell}$ denote the generalized theta graph, which
consists of $\ell$ internally disjoint paths with lengths $k_1,\cdots,
k_{\ell}$, connecting two fixed vertices. We estimate the corresponding
extremal number $\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$. When the lengths of
all paths have the same parity and at most one path has length 1,
$\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$ is $O(n^{1+1/k^\ast})$, where
$2k^\ast$ is the length of the smallest cycle in $\Theta_{k_1,\cdots,k_\ell}$.
We also establish matching lower bound in the particular case of
$\text{ex}(n,\Theta_{3,5,5})$.
