  Bukh and Conlon used random polynomial graphs to give effective lower bounds
on $\mathrm{ex}(n,\mathcal{T}^\ell)$, where $\mathcal{T}^\ell$ is the $\ell$th
power of a balanced rooted tree $T$. We extend their result to give effective
lower bounds on $\mathrm{ex}(G_{n,p},\mathcal{T}^\ell)$, which is the maximum
number of edges in a $\mathcal{T}^\ell$-free subgraph of the random graph
$G_{n,p}$. Analogous bounds for generalized Tur\'an numbers in random graphs
are also proven.
