  We prove that there exists a universal constant $c$ such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.
