There is no nationwide free record search website, but many state court systems and individual local courts offer free online record searches, and more are being added all the time. You can find all of them at CourtReference, which lists court record searches by state and county for every state - along with a lot of other information about courts.
For example, if you want to look up California court records, click on the first source below (Guide to California Courts - Searching Court Case Records) and scroll down the list of counties looking for links to case record searches.
Some states have statewide or multi-county searches. For example, see the second source below (Guide to New York Courts - Searching Court Case Records) which has links to a statewide civil case search, a multi-county criminal record search, and many individual county and local courts' search websites.
From either of these sources, you can select a different state. Or just go to the main site (third source below), which will show you every free case record search website in the US. Choose your state, then go to the "Find court resources by category" drop-down menu and select "Search Court Case Records".