Internet gaming and social media use are prevalent and integral to many people’s lives. However, excessive engagement in either could lead to negative health impacts. This study aimed to investigate relationships between severities of internet gaming disorder (IGD) and problematic social media use (operationalized as social media addiction; SMA) with sleep quality and psychological distress among young adults. A cross-sectional study with snowball sampling was conducted among Hong Kong university students in 2019. All participants (n = 300; mean (SD) age = 20.89 (1.48); 122 males (40.67%)) responded to an online survey that included Chinese versions of the Internet Gaming Disorder Scale-Short Form (IGDS9-SF), Bergen Social Media Addiction Scale (BSMAS), Pittsburgh Sleep Quality Index (PSQI), and Depression Anxiety Stress Scales (DASS-21). Multiple linear regressions demonstrated that IGDS-SF9 scores demonstrated associations with psychological distress measures (standardized coefficient (β) = 0.295 for depression, 0.325 for anxiety, 0.339 for stress, all p < 0.001). BSMAS scores showed similar albeit numerically less robust associations (β = 0.235 for depression, p < 0.001; 0.219 for anxiety, p = 0.001; 0.262 for stress, p < 0.001). BSMAS scores demonstrated associations with poorer sleep quality (β = 0.292; p < 0.001) and IGDS9-SF scores (β = 0.157; p = 0.024) showed a significantly less robust association (p = 0.01 for comparing the two βs). These findings suggest that both severities of IGD and SMA associate with more psychological distress and poorer sleep quality, although the strengths of associations may differ.