We present collapse simulations of 100 M⊙ turbulent cloud cores threaded by a strong magnetic field. During the initial collapse phase, filaments are generated which fragment quickly and form several protostars. Around these protostars Keplerian discs with typical sizes of up to 100 au build up in contrast to previous simulations neglecting turbulence. We examine three mechanisms potentially responsible for lowering the magnetic braking efficiency and therefore allowing for the formation of Keplerian discs. Analysing the condensations in which the discs form, we show that the build-up of Keplerian discs is neither caused by magnetic flux loss due to turbulent reconnection nor by the misalignment of the magnetic field and the angular momentum. It is rather a consequence of the turbulent surroundings of the disc which exhibit no coherent rotation structure while strong local shear flows carry large amounts of angular momentum. We suggest that the ‘magnetic braking catastrophe’, i.e. the formation of sub-Keplerian discs only, is an artefact of the idealized non-turbulent initial conditions and that turbulence provides a natural mechanism to circumvent this problem.