The 21-cm line emission from a 7 × 6 deg2 region east of and adjoining the Small Magellanic Cloud (SMC) has been observed with the Australia Telescope Compact Array and the Parkes telescopes. This region represents the westernmost part of the Magellanic Bridge, a gas-rich tail extending ∼14° to the Large Magellanic Cloud. A rich and complex neutral hydrogen (H i) structure containing shells, bubbles and filaments is revealed. On the larger scale, the H i of the Bridge is organized into two velocity components. This bimodality, which appears to originate in the SMC, converges to a single velocity component within the observed region. A census of shell-like structures suggests a shell population with characteristics similar to that of the SMC. The mean kinematic age of the shells is ∼6 Myr, in agreement with the SMC shell population, but not with ages of OB clusters populating the Magellanic Bridge, which are approximately a factor of 3 older. In general, the projected spatial correlation of Bridge H i shells with OB associations is poor and as such, there does not appear to be a convincing relationship between the positions of OB associations and that of expanding spherical H i structures. This survey has found only one H i shell that has an identifiable association with a known Hα shell. The origin of the expanding structures is therefore generally still uncertain, although current theories regarding their formation include gravitational and pressure instabilities, high-velocity cloud collisions and ram pressure effects.