BRUSSELS—A new European privacy law took effect Friday, causing several major U.S. news websites to suspend access across the region as privacy activists filed complaints and data-protection regulators prepared to brandish their new enforcement powers.

Tronc Inc., publisher of the Los Angeles Times, New York Daily News and other U.S. newspapers, was among those that blocked readers in the European Union from accessing sites, as they scrambled to comply with the sweeping regulation.

“We...