  Given $r$-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 $\rm
ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a
small interval of values of $p=p(n)$ whose length decreases as $\ell$ grows.
  Our main technical contribution is a balanced supersaturation result for
linear even cycles which improves upon previous such results by
Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the
supersaturation result depends on the codegree of some pairs of vertices in the
underlying hypergraph. This approach could be used to prove similar results for
other hypergraphs $H$.
