JEDDAH, Saudi Arabia—Joudi al-Omeri drove in circles. And when cars came in her direction, she swerved. These were electric bumper cars, but in Saudi Arabia, the ride doesn’t always live up to its name.

“I come here to drive,” said Ms. al-Omeri, a 27-year-old homemaker still giddy from the roughly five-minute, mostly crash-free ride in her red-and-green two-seater. “It’s much better than bumping against others,” she adds.

At...