  We prove that there are $O(n)$ tangencies among any set of $n$ red and blue
planar curves in which every pair of curves intersects at most once and no two
curves of the same color intersect. If every pair of curves may intersect more
than once, then it is known that the number of tangencies could be
super-linear. However, we show that a linear upper bound still holds if we
replace tangencies by pairwise disjoint connecting curves that all intersect a
certain face of the arrangement of red and blue curves.
  The latter result has an application for the following problem studied by
Keller, Rok and Smorodinsky [Disc.\ Comput.\ Geom.\ (2020)] in the context of
\emph{conflict-free coloring} of \emph{string graphs}: what is the minimum
number of colors that is always sufficient to color the members of any family
of $n$ \emph{grounded L-shapes} such that among the L-shapes intersected by any
L-shape there is one with a unique color? They showed that $O(\log^3 n)$ colors
are always sufficient and that $\Omega(\log n)$ colors are sometimes necessary.
We improve their upper bound to $O(\log^2 n)$.
