Simonovits's theorem in random graphs

Let $H$ be a graph with $\chi (H)=r+1$. Simonovits’s theorem states that, if $H$ is edge-critical, the unique largest $H$-free subgraph of $K_n$ is its largest $r$-partite subgraph, provided that $n$ is sufficiently large. We show that the same holds with $K_n$ replaced by the binomial random graph $G_{n,p}$ whenever $H$ is also strictly $2$-balanced and $p\ge ({\theta }_H+o(1))n^{-\frac{1}{m_2(H)}}(\log n{)}^{\frac{1}{e_H-1}}$ for some explicit constant ${\theta }_H$, which we believe to be optimal. This (partially) resolves a conjecture of DeMarco and Kahn.