Classes of graphs with superconstant treewidth
I believe that the universal graphs for trees constructed by Chung and Graham 1983 have treewidth $\Theta(\log n)$. Or for a slightly simpler but similar example consider the transitive closures of balanced binary trees.

However, there's a negative result here, too. All the examples you give of interesting graph families are minor-closed, or very closely related to minor-closed families. But a minor-closed graph family either contains all planar graphs (and hence has maximum treewidth $\Theta(\sqrt n)$) or has a forbidden planar minor (and hence has bounded treewidth).