Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free What is the relationship between graphs which have $O(\log n)$ treewidth and $\mathcal{H}$-minor-free graphs? Are graphs which have $O(\log n)$ treewidth $\mathcal{H}$-minor-free? I know that graphs which are $\mathcal{H}$-minor-free have treewidth $O(\sqrt{n})$ but I am concerned more with the relationship between $O(\log n)$ treewidth graphs and $\mathcal{H}$-minor-free graphs.
A proper minor-closed family of graphs has bounded treewidth if and only if it's forbidden minors includes a planar graph. Thus, either the family contains all planar graphs or has bounded treewidth. In the former case the treewidth can be $\Omega(\sqrt{n})$ and in the latter case the treewidth is $O(1)$.