Sorry for the novel...
I'd guess the heavier-than-single-use plastic bags they're selling for $0.10 each if you want bags with handles (at the same store) will cause the same, if not worse issues when they end up in the ocean. The store I'm talking about had the right idea...they were still giving away the single-use bags, but at each entrance to their store also placed a huge bin specifically for recycling them...which people actually DID use. Then the town stepped in and banned the single-use bags, so now that store is giving away paper bags that are too small to use for much of anything (including your groceries - unless you want 10 of them), and selling the thicker, re-usable plastic bags. Granted, they're only charging the ten cents for them, but at what other cost?
If we did some heavy research into those single-use bags, I think we'd find that 1) people aren't re-using them as much as they could (my family always re-used them 3-4 times before they'd tear), 2) that they aren't being disposed of properly (not just tossing them in the trash - actually recycling them), and 3) that there are still a significant number of people who litter instead of actually disposing of trash properly. I'd also venture a guess that those bags aren't as recyclable as they could be at this point.
I'm of the opinion that the single-use plastic bags are just the easiest, most visible target in the trash/recycling mess that exists in the US - especially since China is no longer accepting our trash and recyclables. We won't see real change until we force the businesses that create the things we buy to absorb some of the costs associated with disposal or their products. Planned obsolescence and non-biodegradable packaging are big parts of the problem, but we won't see those kinds of changes until those with the money - the big business CEOs, the politicians in Washington, and the bankers on Wall Street - are able to smell the stench of the trash that is likely the remains of things that made them so wealthy to begin with.
I realize the trash problem isn't just in the US - it's global and at near-emergency status in some parts of the world. The whole situation makes me wonder if the PIXAR film "Wall-e" was a bit more insightful than people think.