We used to use them for this all the time!

Yeah, I used to keep all my grocery bags and use them as garbage bags, at least that got a reuse out of them.

I get doggie poop bags that are allegedly compostable. I never have to worry about the condition of the bag, they come in neat rolls for the dispenser on my leash. They're way handier than disposable shopping bags and cost a few pennies each. I buy a case of them and they last me a few years at a time.

They do seem to be somewhat water soluble. I left some out in the rain one time and they got oddly mushy and weaker, different from regular plastic.