I think you must mean the smaller trash bins sprinkled throughout your house, right? Because a kitchen (main) trash bin is typically too large for little shopping bags? In any case, have you looked into compostable bags? I tried a bunch of different brands and a lot were not very stiff/strong at all until I got a 100pack of stuff that's almost indiscernible from a regular trash bag, except green.

I'm not sure how it works in your house, but in mine the small trash cans throughout rooms are almost entirely filled with compostables like paper towels, tissues, floss, and tampons (I've gotten in arguments about whether floss and tampons will actually compost - that experiment is currently running in our backyard). So you could just Chuck those bags straight into the compost bin, if your city has it. If it doesn't, maybe not a good idea cause the bags will just disintegrate at the landfill and blow trash everywhere.

Really the answer here is convince local government to do recycling and compost collection and make an effort to purchase and cook in such a way that as little goes into the landfill bags as possible. I recently visited my childhood home in Texas and felt super gross when my only option for a can of Coke I had finished was chuck it straight into the bin... I've been thinking a lot lately about how stuff just is accumulating in landfills. Maybe it's my SF instincts screaming "that's a couple square miles of great real estate if it wasn't full of trash!" Lol.