Upper bound on the number of edges of an almost planar bipartite graph

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne6$ and at most $3v-9$ edges for odd $v$ and $v=6$. For all $v\ge4$ examples showing that these bounds are tight are constructed.
In the end of the paper, we discuss a question about drawings of complete bipartite graphs on the plane such that any edge intersects at most one other edge.