While the bloody civil war to oust Mr. Assad grinds on, the various factions arrayed against him are already struggling to reinvent most institutions, including courts, schools, mosques and local administrations. Given that Syria seems to be fracturing into ever-smaller pieces where local warlords hold sway, it is hard to generalize. But there is no question that the Islamic extremists who have gained admiration and respect for their battlefield feats are trying to translate that into authority over society as a whole.

A Shariah Board — as the religious courts are called — recently ordered a secular critic lashed. In mosques, Islamists have pushed moderate clerics off the pulpit — at least once in mid-sermon — replacing them with speakers who harangue worshipers on topics like the evils of hair gel. In one Islamic school, first graders were urged to grow up to become jihadist fighters, said Azzam Khanji, head of education for Aleppo’s Revolutionary Transitional Council, a sort of government in exile for liberated portions of the city.

The court system serves as a prime example of the contest for a postwar Syria. As crime has proliferated after government control vanished in many areas, Syrians clamored for security. Rebel leaders, particularly Islamists, responded by opening dozens of courts.

“It is almost the fashion to have your own courthouse now,” complained Mazen Jumaa of the lawyers association, which monitors the new courts. The association was formed in 2011 to defend young people whose first peaceful antigovernment protests led to the uprising.

The group found that in many cases, fighters lacking training in Shariah, or Islamic law, not to mention civil law, handed down death sentences to government supporters with little or no defense. Commanders of the more secular Free Syrian Army were not much better, the group discovered.