Geometric graphs with no self-intersecting path of length three

Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straight-line edges. It is shown that if G has no self-intersecting path of length 3, then its number of edges is O(nlogn). This result is asymptotically tight. Analogous questions for longer forbidden paths and for graphs drawn by not necessarily straight-line edges are also considered.