Can closing a road make everyone's trip faster?

Yes, in some networks it can, and the reason is that drivers choose routes for themselves, not for everyone. This is Braess's paradox: adding roads to a network can slow overall traffic flow. Arthur Pigou noticed it in 1920, and it was named after the mathematician Dietrich Braess in 1968.

Here is the textbook example. Imagine 4,000 drivers going from Start to End, with two routes, one via point A and one via point B. Each route takes 65 minutes when the drivers split evenly. Now open a new link between A and B that takes almost no time. One driver tries Start–A–B–End and finds it much faster, so others follow. Soon everyone uses the shortcut, and the trip takes 80 minutes instead of 65 for everyone. If drivers agreed not to use the link, or if it were closed, everyone would gain 15 minutes.

Nobody is irrational. Each driver switched because switching was better for them given what the others were doing. The situation is called a Nash equilibrium: nobody has an incentive to change route. But it is a multiplayer prisoner's dilemma, where individually sensible choices make the group worse off.

Real cities provide examples. In Seoul, traffic sped up after the Cheonggye Expressway was removed. In Stuttgart, traffic did not improve until a section of newly built road was closed again. In 1990 the temporary closing of 42nd Street in Manhattan reduced congestion.

It is not magic, though. The paradox appears only in some cases, and other effects, such as trips that simply evaporate, can also explain why closures sometimes help.