Some do, mostly "low cost" stores
And if you look at the places that have introduced the charge, such as M&S, many have adopted a "small bag is free, full size bags are charged" policy as well, presumably in response to negative feedback from customers.
Some other curious data points on this issue, which isn't nearly as black-and-white as it might seem:
For one thing, it turns out that lots of people do "recycle" those "disposable" plastic bags. When Ireland introduced a tax on plastic bags, bin liner purchases increased by 400%.
For another thing, while plastic bags are more environmentally unfriendly than paper bags when discarded, they are more efficient to transport in large numbers, and in practice that inefficiency translates rather directly into increased pollution, greater consumption of non-renewable fuel types for vehicles, and so on. The facts about resources used and pollution generated in manufacture aren't entirely one-sided either.
If the government really wanted to help the environment, they could politely encourage supermarkets to start selling the actually good reusable plastic bags that at least Sainsbury's and Tesco had a few years ago, which were much larger and tougher than the jokes they sell as reusable today (OK, you can reuse them, maybe two or three times before they fall apart). These actually seemed to be quite popular at the time, and we still use some of ours many years later, but the supermarkets that had them all switched to a different and much inferior type after a relatively short time; I don't know why.
In addition, far more environmental good would be done if the government slapped a significant tax on all packaging materials at the source, so that using excessive or unnecessary packaging carried a direct financial penalty. This step alone would almost certainly cut the volume of environmentally unfriendly waste -- meaning waste that can't be recycled or otherwise dealt with other than sending it to landfill -- more than even making all single-use bags of any type completely illegal.
So whenever you see a government official of whatever political affiliation making some claim about helping the environmental by taxing the supply of plastic bags, you should immediately ask what their real agenda is. If they're not also advocating more general restrictions on packaging, and they're not also advocating restricting other environmentally unfriendly practices such as supplying one-time paper bags when reusable bags could be used, then they're probably hiding some ulterior motive and/or capitalizing on some political talking point of the day.