When state officials seek to censor online speech, they're going to use the quickest and easiest method available. For many, copyright takedown notices do the trick. After years of lobbying and increasing pressure from content industries on policymakers and tech companies, sending copyright notices to take media offline is easier than ever.
The copyright law that state actors most often invoke is the Digital Millennium Copyright Act (DMCA). The DMCA was the first major digital copyright law passed in the United States, creating strict procedural rules for how and when a copyright holder can claim that uploaded content infringes on their copyright. US-based tech companies that receive these infringement notices must comply with these rules to receive their safe harbor—the protection they have from being liable for hosting unlawful user content.
The DMCA has become a global tool for censorship, precisely because it was designed to facilitate the removal of online media. The law carries provisions on intermediary liability, among many other strict copyright enforcement rules, which induce websites, Internet service providers, and other such "intermediaries" to remove content that is alleged to be a copyright infringement.