  Given two $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm{ex}(G,
H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the
typical value of $\rm{ex}(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi
random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear
cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open
problem that has been investigated by many researchers. We determine the order
of magnitude of $\rm{ex}\left(G_{n,p}^{(r)}, C_{2\ell}^{(r)}\right)$ for all
$r\geq 4$ and all $\ell\geq 2$ up to polylogarithmic factors for all values of
$p=p(n)$.
  Our proof is based on the container method and uses a balanced
supersaturation result for linear even cycles which improves upon previous such
results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.
