Heavy duty plastic bags might work great if you always keep them in your car.
But I live in NYC where you carry everything by hand -- and people certainly aren't always carrying empty bags with them, the way you might if you had a trunk.
There are two supermarkets I go to where they don't have paper bags, but will charge you $0.25 to $0.45 for a heavy duty plastic bag (two sizes).
I'd say that about a third of the time the person in front of me buys between 1 to 3 of them.
So at least at those locations, the overall usage of plastic has gone way, way up compared to the old thin plastic bags.