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

Let $\text{cC}$ be a set of curves in the plane such that no three curves in $\text{cC}$ intersect at a single point and every pair of curves in $\text{cC}$ 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 $\text{cC}$ that touch each other is $O(\mathrm{\mid }\text{cC}\mathrm{\mid })$. We prove this conjecture for $x$-monotone curves.