The SARS-CoV-2 virus is primarily transmitted through in-person interactions, and so its growth in urban space is a complex function of human mobility behaviors that cannot be adequately explained by standard epidemiological models. Recent studies leveraged fine-grained urban mobility data to accurately model the viral spread, but such data pose privacy concerns and are often difficult to collect, especially in low- and middle-income countries (LMICs). Here, we show that the metapopulation epidemiological model incorporated with a simple gravity mobility model can be sufficient to capture most of the complex epidemic dynamics in urban space, largely reducing the need for empirical mobility data. Extensive experiments on 30 cities in the United States, India and Brazil show that our model consistently reproduces complex, distinctive COVID-19 growth curves with high accuracy. It also provides a theoretical explanation of the emergence of urban "superspreading", where a few high-risk neighborhoods account for most subsequent infections. Furthermore, with the aid of the proposed framework, we can inform mobility-aware travel restrictions to achieve a better balance between social cost and disease prevention, which facilitates sustainable epidemic control and supports the gradual transition to a post-pandemic world.