In the absence of medical treatment and vaccination, individual behaviours are key to controlling the spread of COVID-19. We developed a rapid response monitoring system through a cross-national online survey (the "COVID-19 Health Behavior Survey"). Participant recruitment takes places continuously via Facebook in eight countries (Belgium, France, Germany, Italy, the Netherlands, Spain, the United Kingdom, the United States). The survey collects key information on people's health status, behaviours, and attitudes related to the COVID-19 pandemic. In this paper, we present results based on a total of 71,612 completed questionnaires, collected between March 13-April 19, 2020. We find sex-specific patterns, as women show higher threat perceptions, lower confidence in the healthcare system, and a higher likelihood of adopting preventive behaviours. Our findings also show higher awareness and concern among older respondents. Finally, we find spatio-temporal heterogeneity in threat perception, confidence in organisations, and adoption of preventive behaviours.