DAMASCUS—Demonstrators across Syria pressed their demands for regime change Friday in what some observers called the country's biggest protests here yet.

At least 24 people were killed Friday, human-rights monitors said. Violence continued as President Bashar al-Assad confronts not only mounting public anger but also a growing lawlessness across Syria—ranging from the vigilante groups and Assad supporters who have mounted deadly reprisals on protesters, to a wave of petty crime, uncollected garbage and unauthorized construction....