The oriented Turán 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 Turán number. In this paper, we study the supersaturation phenomenon for oriented Turán problems, and prove oriented versions of the famous Erdős-Simonovits Supersaturation Theorem and Moon-Moser inequality, and supersaturation theorems for tournaments and antidirected complete bipartite graphs.