  We consider the Tur\'an problems of $2$-edge-colored graphs. A
$2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex
set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$
and $E_b$ do not have to be disjoint. The Tur\'an density $\pi(H)$ of $H$ is
defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen
among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as
a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to
measure the edge density of $G_n$. We will determine the Tur\'an densities of
all $2$-edge-colored bipartite graphs. We also give an important application of
our study on the Tur\'an problems of $\{2, 3\}$-hypergraphs.
