KONYA, Turkey—On the humid afternoon after July’s bloody coup attempt, signs of a rift that is redefining this nation’s academia played out in two cities 400 miles apart.

In Istanbul, Nil Mutluer grabbed her 3-year-old daughter and raced with a suitcase toward Turkey’s coast. The former sociology-department chair at the city’s Nisantasi University narrowly escaped the nation’s looming dragnet.

“Authorities...