We perform SPH simulations of the collapse and fragmentation of low-mass cores having different initial levels of turbulence (alpha_turb=0.05,0.10,0.25). We use a new treatment of the energy equation which captures the transport of cooling radiation against opacity due to both dust and gas (including the effects of dust sublimation, molecules, and H^- ions). We also perform comparison simulations using a standard barotropic equation of state. We find that -- when compared with the barotropic equation of state -- our more realistic treatment of the energy equation results in more protostellar objects being formed, and a higher proportion of brown dwarfs; the multiplicity frequency is essentially unchanged, but the multiple systems tend to have shorter periods (by a factor ~3), higher eccentricities, and higher mass ratios. The reason for this is that small fragments are able to cool more effectively with the new treatment, as compared with the barotropic equation of state. We find that the process of fragmentation is often bimodal. The first protostar to form is usually, at the end, the most massive, i.e. the primary. However, frequently a disc-like structure subsequently forms round this primary, and then, once it has accumulated sufficient mass, quickly fragments to produce several secondaries. We believe that this delayed fragmentation of a disc-like structure is likely to be an important source of very low-mass hydrogen-burning stars and brown dwarfs.