Natural (well studied) classes of graphs with treewidth $\Theta(n^\alpha)$ with $1/2 < \alpha < 1$
The intersection graph of interior disjoint balls in $\mathbb{R}^d$, should have treewidth $O(n^{1-1/d})$, if there is justice in the universe (let me think about it - yep - there is). The treewidth should be $\Theta(n^{1-1/d})$. This family of graphs is contained in the family of graphs of low density graphs in $\mathbb{R}^d$. I have a paper (with Kent Quanrud) on low density graphs: http://sarielhp.org/p/14/low_density/low_density.pdf. Sorry for the shameless self promotion, etc.