  This paper studies graphs that have two tree decompositions with the property
that every bag from the first decomposition has a bounded-size intersection
with every bag from the second decomposition. We show that every graph in each
of the following classes has a tree decomposition and a linear-sized path
decomposition with bounded intersections: (1) every proper minor-closed class,
(2) string graphs with a linear number of crossings in a fixed surface, (3)
graphs with linear crossing number in a fixed surface. Here `linear size' means
that the total size of the bags in the path decomposition is $O(n)$ for
$n$-vertex graphs. We then show that every $n$-vertex graph that has a tree
decomposition and a linear-sized path decomposition with bounded intersections
has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on
the crossing number of a graph in terms of its treewidth. Finally, we consider
graph classes that have two path decompositions with bounded intersections.
Trees and outerplanar graphs have this property. But for the next most simple
class, series parallel graphs, we show that no such result holds.
