The COVID-19 pandemic has evolved over time through multiple spatial and temporal dynamics. The varying extent of interactions among different geographical areas can result in a complex pattern of spreading so that influences between these areas can be hard to discern. Here, we use cross-correlation analysis to detect synchronous evolution and potential interinfluences in the time evolution of new COVID-19 cases at the county level in the United States. Our analysis identified two main time periods with distinguishable features in the behavior of correlations. In the first phase, there were few strong correlations that only emerged between urban areas. In the second phase of the epidemic, strong correlations became widespread and there was a clear directionality of influence from urban-to-rural areas. In general, the effect of distance between two counties was much weaker than that of the counties' population. Such analysis can provide possible clues on the evolution of the disease and may identify parts of the country where intervention may be more efficient in limiting the disease spread.