Balanced supersaturation and Turán numbers in random graphs

In a ground-breaking paper solving a conjecture of Erdős on the number of n-vertex graphs not containing a given even cycle, Morris and Saxton [45] made a broad conjecture on so-called balanced supersaturation property of a bipartite graph H. Ferber, McKinley, and Samotij [29] established a weaker version of this conjecture and applied it to derive far-reaching results on the enumeration problem of H-free graphs.In this paper, we show that Morris and Saxton's conjecture holds under a very mild assumption about H, which is widely believed to hold whenever H contains a cycle.We then use our theorem to obtain enumeration results and general upper bounds on the Turán number of a bipartite H in the random graph G(n, p), the latter being the first of its kind.