Topoisomerases relieve the torsional strain in DNA that is built up during replication and transcription. They are vital for cell proliferation and are a target for poisoning by anti-cancer drugs. Type IB topoisomerase (TopIB) forms a protein clamp around the DNA duplex and creates a transient nick that permits removal of supercoils. Using real-time single-molecule observation, we show that TopIB releases supercoils by a swivel mechanism that involves friction between the rotating DNA and the enzyme cavity: that is, the DNA does not freely rotate. Unlike a nicking enzyme, TopIB does not release all the supercoils at once, but it typically does so in multiple steps. The number of supercoils removed per step follows an exponential distribution. The enzyme is found to be torque-sensitive, as the mean number of supercoils per step increases with the torque stored in the DNA. We propose a model for topoisomerization in which the torque drives the DNA rotation over a rugged periodic energy landscape in which the topoisomerase has a small but quantifiable probability to religate the DNA once per turn.