  In a ground-breaking paper solving a conjecture of Erd\H{o}s on the number of
$n$-vertex graphs not containing a given even cycle, Morris and Saxton
\cite{MS} made a broad conjecture on so-called balanced supersaturation
property of a bipartite graph $H$. Ferber, McKinley, and Samotij \cite{FMS}
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 Tur\'an number of a bipartite $H$ in the random
graph $G(n,p)$, the latter being first of its kind.
