There is strong agreement that generative AI should be regulated, but strong disagreement on how to approach regulation. In this position paper, we argue that the debates on generative AI regulation can be informed by evidence on social media regulation. For example, AI companies have faced allegations of political bias similar to the allegations social media companies have faced. We discuss specific policy recommendations based on the evolution of social media and their regulation. These recommendations include investments in (1) efforts to counter bias and perceptions thereof (e.g., via transparency, researcher access, oversight boards, democratic input, research studies), (2) specific areas of regulatory concern (e.g., youth wellbeing, election integrity) and trust and safety, (3) computational social science research, and (4) a more global perspective. Applying lessons learnt from social media regulation to generative AI regulation can save effort and time, and prevent avoidable mistakes.