Remember, we're talking about relative density within a metro area, not how close each metro area is to the others. If you look at cities that were more or less fully developed prior to WW2, like Boston, you find that they're more compact, as opposed to, say, LA or Phoenix.
That said, I don't think it's the only thing making us fat. But I do think it's one contributing factor.
To be fair about the 'single small bridge', it seems like smaller projects suffer proportionately larger delays due to being considered low priority, ridiculously lax work scheduling, etc. Although the financial scales are completely different, it takes my city (Toronto) approximately the same time to build a subway extension as to renovate the bike trail that runs near my house.
Given that sprawl is not a new phenomenon but obesity is I don't see how you can claim this is responsible for making us fat.