This is the literal "gridlock" problem, which can lock up a road system. The US had that problem in some cities until New York City introduced "Don't Block the Box" around 1980. It took a PR campaign, and, some ticketing, and painting a big box with diagonal lines across some intersections, but eventually drivers got the idea to not enter the intersection until they had a way out of it. San Francisco is starting to have gridlock again, and a new "Don't Block the Box" campaign was started in 2014.