We present collapse simulations of strongly magnetised , turbulent molecular cloud cores with masses ranging from 2.6 to 1000 M _ { \sun } in order to study the influence of the initial conditions on the turbulence-induced disc formation mechanism proposed recently by Seifried et al . ( 66 ) . We find that Keplerian discs are formed in all cases independently of the core mass , the strength of turbulence , or the presence of global rotation . The discs appear within a few kyr after the formation of the protostar , are 50 – 150 AU in size , and have masses between 0.05 and a few 0.1 M _ { \sun } . During the formation of the discs the mass-to-flux ratio stays well below the critical value of 10 for Keplerian disc formation . Hence , flux-loss alone can not explain the formation of Keplerian discs . The formation of rotationally supported discs at such early phases is rather due to the disordered magnetic field structure and due to turbulent motions in the surroundings of the discs , two effects lowering the classical magnetic braking efficiency . Binary systems occurring in the discs are mainly formed via the disc capturing mechanism rather than via disc fragmentation , which is largely suppressed by the presence of magnetic fields .