Catalytic introduction of negative supercoils into a DNA molecule or region thereof. In bacteria, negative supercoils are only introduced by DNA gyrase, a type II topoisomerase, but not all DNA gyrases are capable of introducing supercoils. In bacteria, the level of supercoiling varies widely between species and has been characterized properly in only a handful of organisms. The best characterized enzyme, from E.coli, is exceptionally proficient at supercoiling and this ability is not representative of all bacteria.