Effectively designing and evaluating public health responses to the ongoing COVID-19 pandemic requires accurate estimation of the weekly incidence of COVID-19. Unfortunately, a lack of systematic testing across the United States (US) due to equipment shortages and varying testing strategies has hindered the usefulness of the reported positive COVID-19 case counts. We introduce three complementary approaches to estimate the cumulative incidence of symptomatic COVID-19 during the early outbreak in each state in the US as well as in New York City, using a combination of excess influenza-like illness reports, COVID-19 test statistics, and COVID-19 mortality reports. Instead of relying on an estimate from a single data source or method that may be biased, we provide multiple estimates, each relying on different assumptions and data sources. Across our three approaches, there is a consistent conclusion that estimated state-level COVID-19 symptomatic case counts from March 1 to April 4, 2020 varied from 5 to 50 times greater than the official positive test counts. Nationally, our estimates of COVID-19 symptomatic cases in the US as of April 4 have a likely range of 2.2 to 5.1 million cases, with possibly as high as 8.1 million cases, up to 26 times greater than the cumulative confirmed cases of about 311,000. Extending our method to May 16, 2020, we estimate that cumulative symptomatic incidence ranges from 6.0 to 12.2 million, which compares with 1.5 million positive test counts. Our approaches demonstrate the value of leveraging existing influenza-like-illness surveillance systems during the flu season for measuring the burden of new diseases that share symptoms with influenza-like-illnesses. Our methods may prove useful in assessing the burden of COVID-19 during upcoming flu seasons in the US and other countries with comparable influenza surveillance systems.