As large language models (LLMs) are increasingly used to address health-related queries, it is crucial that they support their conclusions with credible references. While models can cite sources, the extent to which these support claims remains unclear. To address this gap, we introduce SourceCheckup, an automated agent-based pipeline that evaluates the relevance and supportiveness of sources in LLM responses. We evaluate seven popular LLMs on a dataset of 800 questions and 58,000 pairs of statements and sources on data that represent common medical queries. Our findings reveal that between 50% and 90% of LLM responses are not fully supported, and sometimes contradicted, by the sources they cite. Even for GPT-4o with Web Search, approximately 30% of individual statements are unsupported, and nearly half of its responses are not fully supported. Independent assessments by doctors further validate these results. Our research underscores significant limitations in current LLMs to produce trustworthy medical references.