Europe is rapidly moving away from primary reliance on the criminal law for dealing with online ‘extreme speech’ towards use of notice-and-takedown procedures, as under the EU’s ‘voluntary’ Hate Speech Code and the German NetzDG law. The scale of these operations dwarfs the small number of prosecutions in European states, while their lightning-quick operation (often within 24 hours) make the traditional criminal justice system look impossibly ponderous. The new Regulation on tackling terrorism content online will move the EU to a compulsory system, backed by heavy fines. This paper argues that the EU approach indiscriminately lumps together wholly different types of ‘terrorism-related material’ that, in free speech terms, have wildly different value. But it challenges the notion that the Regulation is simply a retrograde step in civil liberties terms, contending that transparency, due process, access to legal speech rights and possibilities for legal challenge will all be enhanced.