I have a little stash of paper and plastic bags that I dust out and reuse many many times at the bulk counter. Carrying glass in to the store would just be a nightmare for me, with a running toddler. But I can get probably 20 uses (or more) out of a paper bag, and maybe twice that out of a little plastic bag. So it's not 100% sustainable, but it goes a lot farther than originally designed for, and it's still lightweight. This is all for small/tiny/fine items like flour, falafel, sugars, herbs, etc.
I'm crocheting some little stringy bags for produce, which are v. lightweight and save even more. I'm color-coordinating them -- several green ones, but also orange for carrots or oranges, red for tomatoes/apples/peppers, yellow for bananas/grapefruit, purple for grapes/onions, brown for potatoes, white for mushrooms/daikon. It's fun!