It takes a lot of energy to make a paper bag. The paper industry is a huge energy consumer and polluter. Sainsbury's reusable veggie bags are made from recycled plastic bottles so that hasn't removed plastic from the equation. But, we have to start somewhere, as BlueBelle points out.
We used to buy our veg loose from the greengrocer and had to take our own bags to carry it home. I still do that, even in the supermarket, it it's offered loose, but the smaller, local stores don't have room for large displays of loose produce. Unfortunately, I rarely remember to remove packaging and leave it behind in the shop. You can do that, if you think on.
In Northumberland, the household recycling collection is restricted to plastic bottles, cans, paper and card. Plastic wrapping, plastic trays from meat and veg, yogurt pots and food waste etc. have to go in the general waste (unless you compost veg peelings, as I do). There are local hubs for glass bottles and textiles. Our village tetrapack container has recently been removed. Anything else must be taken (in my case) ten miles to the nearest recycling centre. My daughter and her husband are scandalised by Northumberland's poor performance compared with Bristol, where they accept almost everything for recycling.
What happens to it then, mind you, is another story, another scandal when you read about our waste being shipped across the world, where they dump it in landfill, in the sea, or pollute the air by burning it. However, sometimes they send it straight back because they find hazardous waste concealed in other, recyclable stuff. I've considered deleting this since it's quite depressing....?