Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The random Turán number $\mathrm{ex}(G^r_{n,p},\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erdös-Rényi random $r$-graph with parameter $p$. Let $C^r_{\ell}$ denote the $r$-uniform linear cycle of length $\ell$. For $p\ge n^{-r+2+o(1)}$, Mubayi and Yepremyan showed that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})\le\max\{p^{\frac{1}{2\ell-1}}n^{1+\frac{r-1}{2\ell-1}+o(1)},pn^{r-1+o(1)}\}$. This upper bound is not tight when $p\le n^{-r+2+\frac{1}{2\ell-2}+o(1)}$. In this paper, we close the gap for $r\ge 4$. More precisely, we show that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})=Θ(pn^{r-1})$ when $p\ge n^{-r+2+\frac{1}{2\ell-1}+o(1)}$. Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For $r=3$, we significantly improve Mubayi and Yepremyan's upper bound. Moreover, we give reasonably good upper bounds for the random Turán numbers of Berge even cycles, which improve previous results of Spiro and Verstraëte.