  A circle graph is an intersection graph of a set of chords of a circle. We
describe the unavoidable induced subgraphs of circle graphs with large
treewidth. This includes examples that are far from the `usual suspects'. Our
results imply that treewidth and Hadwiger number are linearly tied on the class
of circle graphs, and that the unavoidable induced subgraphs of a
vertex-minor-closed class with large treewidth are the usual suspects if and
only if the class has bounded rank-width. Using the same tools, we also study
the treewidth of graphs $G$ that have a circular drawing whose crossing graph
is well-behaved in some way. In this setting, we show that if the crossing
graph is $K_t$-minor-free, then $G$ has treewidth at most $12t-23$ and has no
$K_{2,4t}$-topological minor. On the other hand, we show that there are graphs
with arbitrarily large Hadwiger number that have circular drawings whose
crossing graphs are $2$-degenerate.
