Girth and treewidth

The length of the shortest cycle in a graph G is called the girth of G. In particular, we show that if G has girth at least g and average degree at least d, then tw(G)=Ω(1g+1 (d−1)⌊(g−1)/2⌋). In view of a famous conjecture regarding the existence of graphs with girth g, minimum degree δ and having at most c(δ−1)⌊(g−1)/2⌋ vertices (for some constant c), this lower bound seems to be almost tight (but for a multiplicative factor of g+1).