Disrupting the emergence and evolution of potentially violent online extremist movements is a crucial challenge. Extremism research has analyzed such movements in detail, focusing on individual- and movement-level characteristics. But are there system-level commonalities in the ways these movements emerge and grow? Here we compare the growth of the Boogaloos, a new and increasingly prominent U.S. extremist movement, to the growth of online support for ISIS, a militant, terrorist organization based in the Middle East that follows a radical version of Islam. We show that the early dynamics of these two online movements follow the same mathematical order despite their stark ideological, geographical, and cultural differences. The evolution of both movements, across scales, follows a single shockwave equation that accounts for heterogeneity in online interactions. These scientific properties suggest specific policies to address online extremism and radicalization. We show how actions by social media platforms could disrupt the onset and ‘flatten the curve’ of such online extremism by nudging its collective chemistry. Our results provide a system-level understanding of the emergence of extremist movements that yields fresh insight into their evolution and possible interventions to limit their growth.