Social media have been central in informing people about the COVID-19 pandemic. They influence the ways in which information is perceived, communicated and shared online, especially with physical distancing measures in place. While these technologies have given people the opportunity to contribute to public discussions about COVID-19, the narratives disseminated on social media have also been characterised by uncertainty, disagreement, false and misleading advice. Global technology companies have responded to these concerns by introducing new content moderation policies based on the concept of harm to tackle the spread of misinformation and disinformation online. In this essay, we examine some of the key challenges in implementing these policies in real time and at scale, calling for more transparent and nuanced content moderation strategies to increase public trust and the quality of information about the pandemic consumed online.