  The random Tur\'an number $\mathrm{ex}(G_{n,p}^r,F)$ is the maximum number of
edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$.
We prove general results which (informally) shows that if $F$ is an
$r_0$-graph, then upper bounds for $\mathrm{ex}(G_{n,p}^{r_0},F)$ can be lifted
into upper bounds for $\mathrm{ex}(G_{n,p}^{r},F^{(r)})$ where $F^{(r)}$ is the
$r$-uniform expansion of $F$, i.e.\ the hypergraph obtained from $F$ by
inserting $r-r_0$ distinct vertices into each edge of $F$. These results unify
and generalize most known upper bounds for random Tur\'an numbers of degenerate
hypergraphs of uniformity at least 3, and also provide new tight bounds for the
random Tur\'an numbers of expansions of theta graphs and complete bipartite
graphs.
