CAIRO—In Egypt, the websites of the Huffington Post and Human Rights Watch aren’t available to most internet users. The Turkish government blocks more than 100,000 sites, including Wikipedia. In Saudi Arabia, a range of news sites linked to rivals Qatar and Iran are off limits.

“My first thought was, ‘Welcome to China,’” said a banker in Cairo, recalling his attempt to access Mada Masr, Egypt’s leading independent news organization, which has been blocked since June 2017. He asked to have his name withheld for fear of government...