The campaign started with the cyber equivalent of a massive airstrike: law-enforcement agencies from eight countries, moving in unison to smash two of the main propaganda organs of the Islamic State. In the two-day operation in April, police seized computers and network servers across Europe and North America and blocked Internet portals used by the terrorist group’s radio broadcaster, al-Bayan, and its official news agency, Amaq. Yet, less than a week later, Amaq suddenly reappeared at a different Web address, forcing the governments to pounce again. Then it surfaced a third time. And a fourth. Today, more than four months after the European police agency Europol began the initiative, the struggle to silence the Islamic State’s communications flagships has shifted from shock-and-awe to something resembling trench warfare. Within weeks of the Europol operation, more Amaq videos started turning up on Telegram, and then were reposted on other social media platforms, including YouTube and Facebook. A study released last month by the Counter Extremism Project, a New York-based nonprofit, found that a quarter of Islamic State videos uploaded onto YouTube remained accessible for at least two hours before being discovered and taken down, according to the study, which analyzed content over three months last spring. On other platforms, the links remained active for days or even weeks, potentially allowing new videos to be downloaded and shared thousands of times.