Performative and possibly even counterproductive. The other bags you can use will usually require a larger consumption of fossil fuels consumed over their likely use lifetime. Paper bags are more energy intensive (including burning fossil fuels for energy). Cloth bags are much, much more energy intensive, you need to use each one hundreds of times for it to balance out.

If the concern is pollution from plastics, grocery bags are a very minor contributor. Most can simply be reused as trash bags or trashed/incinerated. It's tires and runoff from freeways that are the elephant in the room but addressing that requires building a real mass transit system.