Did Lockdowns Reduce COVID-19 Transmission?

Americans have been debating the effectiveness of COVID-19 lockdowns for months now. To what extent did broad business closure and stay-at-home orders—as opposed to narrower interventions and voluntary precautions—reduce virus transmission, thereby avoiding hospital crises, buying time to develop testing and tracing capacity, and preventing deaths that otherwise would have occurred? Conversely, to what extent can loosening those restrictions be expected to increase transmission, leading to more infections and deaths?