  Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an
$F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$. Building on
recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on
$\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko. This connection
between Sidorenko's conjecture and random Tur\'an problems gives new lower
bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko, and further
allows us to bound how "far" from Sidorenko an $r$-graph $F$ is whenever upper
bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known.
