BEIRUT — New chaos engulfed Syria’s civil war Monday as Palestinian supporters and opponents of the embattled regime were swept up in intense fighting in Damascus, while rival rebel groups clashed over control of a Turkish border crossing.

The rare infighting — accompanied by car bombs, airstrikes, and artillery shells that killed or maimed dozens of people — heightened fears that if President Bashar Assad of Syria falls, the disparate factions battling the regime will turn on each other.