Maximum number of intersections between a quadrilateral and a pentagon

$16$ is the maximum if one proves that a line cannot meet the boundary of a $(2n+1)$-gon at more than $2n$ points. And this is a consequence of the Jordan curve theorem: each time we cross the boundary of a polygon we go from the exterior to the interior or the opposite. If we start at the exterior and we end at the exterior, we have an even number of crossings.

A line segment can only intersect four sides of a pentagon. The best you can do is put three vertices on one side of the line segment and two on the other. When you connect the vertices of the pentagon you have to connect two of the three on one side of the line segment and that side will not intersect. Your example has each of the four sides of the quadrilateral intersecting four sides of the pentagon, so this is maximal.