Observations of turbulent velocity dispersions in the H i component of galactic discs show a characteristic floor in galaxies with low star formation rates and within individual galaxies the dispersion profiles decline with radius. We carry out several high-resolution adaptive mesh simulations of gaseous discs embedded within dark matter haloes to explore the roles of cooling, star formation, feedback, shearing motions and baryon fraction in driving turbulent motions. In all simulations the disc slowly cools until gravitational and thermal instabilities give rise to a multiphase medium in which a large population of dense self-gravitating cold clouds are embedded within a warm gaseous phase that forms through shock heating. The diffuse gas is highly turbulent and is an outcome of large-scale driving of global non-axisymmetric modes as well as cloud–cloud tidal interactions and merging. At low star formation rates these processes alone can explain the observed H i velocity dispersion profiles and the characteristic value of ∼10 km s−1 observed within a wide range of disc galaxies. Supernovae feedback creates a significant hot gaseous phase and is an important driver of turbulence in galaxies with a star formation rate per unit area ≳10−3 M⊙ yr−1 kpc−2 .