Unverified information concerning COVID-19 can affect mental health. Understanding perceived trust in information sources and associated mental health outcomes during the COVID-19 pandemic is vital to ensure ongoing media coverage of the crisis does not exacerbate mental health impacts. A number of studies have been conducted in other parts of the world to determine associations between information exposure relating to COVID-19 and mental health. However, the mechanism by which trust in information sources may affect mental health is not fully explained in the developing country context. To address this issue, the present study examined associations between perceived trust in three sources of information concerning COVID-19 and anxiety/stress with the mediating effects of COVID-19 stress in Bangladesh. An online cross-sectional study was conducted with 744 Bangladeshi adults between 17 April and 1 May 2020. Perceived trust in traditional, social, and health media for COVID-19 information, demographics, frontline service status, COVID-19-related stressors, anxiety (GAD-7), and stress (PSS-4) were assessed via self-report. Linear regression tested for associations between perceived trust and mental health. Mediation analyses investigated whether COVID-19-related stressors affected perceived trust and mental health associations. In fully adjusted models, more trust in social media was associated with more anxiety (B = 0.03, CI = 0.27–0.97) and stress (B = 0.01, CI = −0.34–0.47), while more trust in traditional media was associated with more anxiety (B = 0.09, CI = 0.17–2.26) but less stress (B = −0.08, CI = −0.89–0.03). Mediation analyses showed that COVID-19-related stressors partially explained associations between perceived trust and anxiety. These findings suggest that trusting social media to provide accurate COVID-19 information may exacerbate poor mental health. These findings also indicate that trusting traditional media (i.e., television, radio, and the newspaper) may have stress-buffering effects. We recommend that responsible authorities call attention to concerns about the trustworthiness of social media as well as broadcast positive and authentic news in traditional media outcomes based on these results.