The intensification of the hydrological cycle has altered the spatiotemporal redistribution of terrestrial water storage (TWS). The long-term evolutionary mechanism controlling global TWS droughts and their bivariate risks (i.e., drought and severity) remain uncertain. Using machine learning-based TWS reconstructions, we explored the drivers, changes, and impacts of TWS droughts during 1940-2022 at the global scale. We developed a machine learning framework to detect the dominant climate/vegetation factors governing TWS. During 1940-1970, precipitation and vapor pressure deficit were the primary factors influencing TWS; however, the leaf area index was the dominant factor during 1992-2022. We evaluated past changes in drought frequency, duration, severity, and intensity, and found a substantial intensification tendency in most land areas. Subsequently, we evaluated the bivariate risks by combing a copula-based modeling approach and the most likely realization method, revealing a fivefold intensification over most regions. Changes in the marginal distributions of duration and severity accounted for 40-60% of the overall changes in bivariate drought risk, while the contribution from their dependence varied globally. Approximately 80-90% of the global population and gross domestic product were exposed to increasing bivariate drought risk, indicating the need to improve the adaptivity of society and ecosystems to climate change.