Bags don’t fly free with Southwest any longer. In March, Dallas-based Southwest Airlines announced a major change to its longstanding bags-fly-free policy: checked bags, for the most part, would no longer fly free.

Starting Wednesday, the airline will charge customers $35 for one checked bag and $45 for a second, unless they hold the airline’s credit card or have elite status.

The airline estimated in September that charging bag fees would bring in about $1.5 billion a year but cost the airline $1.8 billion in lost business from customers who chose to fly Southwest because...