On the supersaturation of oriented Tur\'an problems

The oriented Tur\'{a}n number of a given oriented graph $\overrightarrow{F}$, denoted by $\exo(n,\overrightarrow{F})$, is the largest number of arcs in $n$-vertex $\overrightarrow{F}$-free oriented graphs. This parameter could be seen as a natural oriented version of the classical Tur\'{a}n number. In this paper, we study the supersaturation phenomenon for oriented Tur\'{a}n problems, and prove oriented versions of the famous Erd\H{o}s-Simonovits Supersaturation Theorem and Moon-Moser inequality, and supersaturation theorems for transitive tournaments and antidirected complete bipartite graphs.