Plastic bags are a good long-term alternative to paper bags. They don't break easily - I have been using the same bag for 5-6 years for Seven-11 runs - and they are super lightweight. The problem, of course, is how people dispose of them plus the scarcity of broken bag recyclers.
As I have yet to find the end-of-life of my trusty plastic bag, I still recommend them. Honestly, paper bags are single use only in the rain so just take a plastic bag.