During the COVID-19 pandemic, scientific knowledge evolved rapidly, accompanied by a surge of misinformation, labelled an infodemic by the WHO. In this context, we study the interaction between science and misinformation on Twitter (now X) using a database of ~407M COVID-19-related tweets. We classify URL reliability with Media Bias/Fact Check and used Altmetric data to identify scientific publications. We find that among ~1.2M users who shared science, 45% also shared unreliable content. Scientific papers circulated by these users were more often preprints, slightly more likely to be retracted, less cited, and published in lower-impact journals. Our findings indicate misinformation is not driven by a lack of exposure to science but instead raise critical questions about open science practices, particularly the role of preprints in amplifying misleading narratives. Our results underscore the importance of proactive scientific engagement on social media in countering misinformation and reinforcing trust in science during global crises.