  Let $V_n$ be a set of $n$ points in the plane and let $x \notin V_n$. An
$x$-loop is a continuous closed curve not containing any point of $V_n$. We say
that two $x$-loops are non-homotopic if they cannot be transformed continuously
into each other without passing through a point of $V_n$. For $n=2$, we give an
upper bound $e^{O\left(\sqrt{k}\right)}$ on the maximum size of a family of
pairwise non-homotopic $x$-loops such that every loop has fewer than $k$
self-intersections and any two loops have fewer than $k$ intersections. The
exponent $O\big(\sqrt{k}\big)$ is asymptotically tight. The previous upper
bound bound $2^{(2k)^4}$ was proved by Pach, Tardos, and T\'oth [Graph Drawing
2020]. We prove the above result by proving the asymptotic upper bound
$e^{O\left(\sqrt{k}\right)}$ for a similar problem when $x \in V_n$, and by
proving a close relation between the two problems.
