The plastic bags are made from can be recycled. If you're determined you can recycle almost anything. However, the bags are a total nightmare for automated or semi-automated sorting of mixed recycling.
If you give me 10 trucks per day of mixed soda cans and newspapers, I can build a factory the size of a small building that separates and processes them into bales of crushed steel cans, bales of crushed aluminium cans, and bales of newspaper, all three have recycling markets and I can sell them.
If we add "some plastic bags" to those truckloads my machine is now a nightmare. The bags will tangle with anything, and they might more or less at random "hop" over or fall into holes sized to match objects, the machine that sorts alumium cans can't handle them, the machine that bales newspapers can't handle them, everywhere is now "also plastic bags", and all my customers don't want plastic bags, their specifications are clear, a little bit of Pepsi or Budweiser in the bale of steel cans is tolerable, a plastic bag is not; One cardboard loo roll mixed in with the newspaper is tolerable, a plastic bag is not; And so on.
I visited a plant in Central London that handles their area's recycling - very noisy, very interesting, they said if visitors took away just one single idea from their visit that idea should be "Don't put any fucking plastic bags in the general recycling".