We present the kinematic results from our ARGOS spectroscopic survey of the Galactic bulge of the Milky Way. Our aim is to understand the formation of the Galactic bulge. We examine the kinematics of about 17 400 stars in the bulge located within 3.5 kpc of the Galactic Centre, identified from the 28 000 star ARGOS survey. We aim to determine if the formation of the bulge has been internally driven from disc instabilities as suggested by its boxy shape, or if mergers have played a significant role as expected from lambda cold dark matter simulations. From our velocity measurements across latitudes b = −5°, − 7 5 and −10° we find the bulge to be a cylindrically rotating system that transitions smoothly out into the disc. From observations of 3 fields at b = +10, the kinematics of the bulge show North-South symmetry about the major axis. Within the bulge, we find a kinematically distinct metal-poor population (Fe/H < −1.0) that is not rotating cylindrically. The 5 per cent of our stars with Fe/H < −1.0 are a slowly rotating spheroidal population, which we believe are stars of the metal-weak thick disc and halo which presently lie in the inner Galaxy. The kinematics of the two bulge components that we identified in ARGOS Paper III (mean Fe/H −0.25 and Fe/H +0.15, respectively) demonstrate that they are likely to share a common formation origin and are distinct from the more metal-poor populations of the thick disc and halo which are co-located inside the bulge. We do not exclude an underlying merger generated bulge component but our results favour bulge formation from instabilities in the early thin disc.