Most plastic bags can be recycled by returning them to the retail location where you received them. In the United States, large retail stores like Walmart and Target and many grocery chain stores have receptacles in the store and they will accept clean and dry plastic bags for recycling. The collected bags are then sent to a recycling facility that can process them properly.
If thin plastic bags are included in general recycling bins collected by curb side recycling companies, they make the sorting process more difficult and they can end up clogging machinery. If you use reusable shopping bags, you can avoid having to use plastic.