Tight bound on the number of intersections between a line and a triangulation
Place $T$ triangles next to each other such that the line $l$ intersects them all. Observe how the number of intersections is $T+1$. Now substitute $T$ with the upper bound on the number of triangles for any triangulation that you mentioned, and we obtain $2n-5+1=2n-4$ as an upper bound on the number of intersections. The example provided in the question shows that the bound is tight.