With the continued spread of coronavirus, the task of forecasting distinctive COVID-19 growth curves in different cities, which remain inadequately explained by standard epidemiological models, is critical for medical supply and treatment. Predictions must take into account non-pharmaceutical interventions to slow the spread of coronavirus, including stay-at-home orders, social distancing, quarantine and compulsory mask-wearing, leading to reductions in intra-city mobility and viral transmission. Moreover, recent work associating coronavirus with human mobility and detailed movement data suggest the need to consider urban mobility in disease forecasts. Here we show that by incorporating intra-city mobility and policy adoption into a novel metapopulation SEIR model, we can accurately predict complex COVID-19 growth patterns in U.S. cities ($R^2$ = 0.990). Estimated mobility change due to policy interventions is consistent with empirical observation from Apple Mobility Trends Reports (Pearsons R = 0.872), suggesting the utility of model-based predictions where data are limited. Our model also reproduces urban superspreading, where a few neighborhoods account for most secondary infections across urban space, arising from uneven neighborhood populations and heightened intra-city churn in popular neighborhoods. Therefore, our model can facilitate location-aware mobility reduction policy that more effectively mitigates disease transmission at similar social cost. Finally, we demonstrate our model can serve as a fine-grained analytic and simulation framework that informs the design of rational non-pharmaceutical interventions policies.