Maximum Treewidth of a Graphs with $m$ Edges
There are bounded degree (in fact even cubic i.e. 3-regular) expanders on $m$ vertices with treewidth at least $\epsilon \cdot m$, for some constant $\epsilon &gt; 0$. See Section 2 in:

Martin Grohe, Dániel Marx:
On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99(1): 218-228 (2009)

Basically, they prove something much stronger: large vertex expansion is the only reason on why the treewidth can be as much as linear (in the number of vertices, but bounded degree graphs have linear number of edges).