Crossings between non-homotopic edges

A multigraph drawn in the plane is called non-homotopic if no pair of its edges connecting same vertices can be continuously transformed into each other without passing through a vertex, and loop shrunk to end-vertex way. Edges are allowed intersect themselves. It easy see that on n>1 have arbitrarily many edges. We prove number crossings between with n m>4n larger than cm2n for some constant c>0, this bound tight up polylogarithmic factor. also show lower not asymptotically sharp as fixed m tends infinity.