  We prove that the number of tangencies between the members of two families,
each of which consists of $n$ pairwise disjoint curves, can be as large as
$\Omega(n^{4/3})$. We show that from a conjecture about forbidden $0$-$1$
matrices it would follow that this bound is sharp for doubly-grounded families.
We also show that if the curves are required to be $x$-monotone, then the
maximum number of tangencies is $\Theta(n\log n)$, which improves a result by
Pach, Suk, and Treml. Finally, we also improve the best known bound on the
number of tangencies between the members of a family of at most
$t$-intersecting curves.
