You can try this query, which I simplified and generalized from a more specific example on the wiki doc.
It search for all ways with highway tags for roads, and then look for the nodes connecting at least three of those. The request runs fine on a small area, not sure for wider. It looks for intersections in the bounding box based on the current map view. You can look in an area (like a county or something) instead of the bbox, you’ll find examples on the wiki, or using the assistant.
The list of highway= values is on the relevant page. You probably want to exclude service roads as it add a lot of data.