Princeton Library's Health Data Guide is the best starting point for health data and statistics
Below are some highlights and additional sources.
International
Demographic and Health Surveys - Published by the DHS Program and USAID. Collects, analyzes and disseminates data and reports about population, health, HIV, and nutrition on over 90 countries.
OECD Health DataExamines national health systems from 1960 forward for OECD member countries in a general, demographic, economic, and social context.
Global Health Observatory (formerly WHOSIS)Guide to health and health-related epidemiological and statistical information available from the World Health Organization. For other information from the World Health Organization, see their website.
World Development Indicators Includes annual economic, social, educational, environmental and health data from many of the World Bank's major statistical publications. 1960+
Health Poll Database Provides health-related U.S. survey results, covering eighty years of national polling. Searchable questions and results, demographic crosstabs, and trends are available on every topic related to health, from social determinants and influences on health to insurance, costs, health-care utilization and COVID.
Kaiser Family Foundation: State Health Facts Health data for all 50 states, the District of Columbia, the United States, counties, territories, and other geographies, from a variety of public and private sources, including KFF reports, public websites, government surveys and reports, and private organizations.
The Anne E. Casey Foundation: Kids Count Data Center - Statistics and data about child well-being and health by state, county, city and Congressional district. Covers birth outcomes, dental health, vital statistics, health insurance, mental health, immigrant children. Data drawn from government agencies and health policy organizations.
PeriStats - Perinatal statistics developed by the March of Dimes Perinatal Data Center. Maternal and infant health-related data at the U.S., state, county, and city level. Also includes reports.