  For a graph $G$ and a hereditary property $\mathcal{P}$, let
$\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of
$G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial
hereditary property $\mathcal{P}$ such that $L \notin \mathcal{P}$ for some
bipartite graph $L$ and for every fixed $p \in (0,1)$ we have
\[\text{ex}(G(n,p),\mathcal{P}) \le n^{2-\varepsilon}\] with high probability,
for some constant $\varepsilon = \varepsilon(\mathcal{P})>0$. This answers a
question of Alon, Krivelevich and Samotij.
