  We investigate natural Tur\'an problems for mixed graphs, generalizations of
graphs where edges can be either directed or undirected. We study a natural
\textit{Tur\'an density coefficient} that measures how large a fraction of
directed edges an $F$-free mixed graph can have; we establish an analogue of
the Erd\H{o}s-Stone-Simonovits theorem and give a variational characterization
of the Tur\'an density coefficient of any mixed graph (along with an associated
extremal $F$-free family). This characterization enables us to highlight an
important divergence between classical extremal numbers and the Tur\'an density
coefficient. We show that Tur\'an density coefficients can be irrational, but
are always algebraic; for every positive integer $k$, we construct a family of
mixed graphs whose Tur\'an density coefficient has algebraic degree $k$.
