The manufacture of plastic bags produces much less carbon emissions than paper bags. Consider the costs of logging, transportation of wood, the manufacture process which uses a ton of water, the transportation of paper which is heavier than plastic which means higher fuel costs, etc. And also consider that most trees we cut down from paper come directly from farms which often require irrigation or items like fertilizer (which have carbon costs). Although not every tree farm uses that, some are more "natural growth"

Plastic bags tend to be more durable and re-usable than paper bags. Unfortunately most people don't re-use either.

Of course, the main issue is the fact that they take hundreds of years to decompose and end up everywhere. Also, plastics come directly from petrochemicals which are a finite resource. There are ways to create plastic from renewable oils, although that raises the carbon emissions significantly.

I think this is an excellent example to give people to illustrate that a lot of times, the choices we make as a society about simple things can be counter-intuitive. Often times, we're making decisions about what bad thing we want less. Do we want plastic building up in landfills and oceans, or do we want the global temperature to stop rising?

Of course, these aren't the only two options and it's not a 1 to 1 linear relationship. But it's an interesting example.