On the number of tangencies among 1-intersecting x-monotone curves

Let C be a set of curves in the plane such that no three curves in C intersect at a single point and every pair of curves in C intersect at exactly one point which is either a crossing or a touching point. János Pach conjectured that the number of pairs of curves in C that touch each other is O(|C|). We prove this conjecture for x-monotone curves.