  The treewidth is a structural parameter that measures the tree-likeness of a
graph. Many algorithmic and combinatorial results are expressed in terms of the
treewidth. In this paper, we study the treewidth of outer $k$-planar graphs,
that is, graphs that admit a straight-line drawing where all the vertices lie
on a circle, and every edge is crossed by at most $k$ other edges.
  Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar
graph has treewidth at most $3k + 11$ using so-called planar decompositions,
and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer
$1$-planar graphs is at most $3$, which is tight.
  In this paper, we improve the general upper bound to $1.5k + 2$ and give a
tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that,
for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$.
Our new bound immediately implies a better bound on the cop number, which
answers an open question of Durocher et al. [GD 2023] in the affirmative.
  Our treewidth bound relies on a new and simple triangulation method for outer
$k$-planar graphs that yields few crossings with graph edges per edge of the
triangulation. Our method also enables us to obtain a tight upper bound of $k +
2$ for the separation number of outer $k$-planar graphs, improving an upper
bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer
min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we
achieve smaller improvements.
