Context. Theoretical and numerical studies of star formation have shown that a magnetic field can greatly influence both disk formation and its fragmentation, with even relatively low magnetic field strengths being able to prevent these processes. However, very few studies have investigated the combined effects of magnetic field and turbulence. Aims. We study the collapse of turbulent, magnetized prestellar cores, focusing on the effects of magnetic diffusion, and misalignment between rotation axis and magnetic field, on the formation of disks, fragmentation, and the generation of outflows. Methods. We performed three-dimensional, adaptive-mesh, numerical simulations of magnetically super-critical collapsing dense cores of 5 M⊙ using the magneto-hydrodynamic code Ramses. A turbulent velocity field is imposed as initial conditions, characterized by a Kolmogorov power spectrum. Different levels of turbulence (a laminar case, as well as subsonic and supersonic cases) and magnetization (from weak to strong magnetization) are investigated, as are three realizations for the turbulent velocity field. Results. The turbulent velocity field imposed as initial conditions contains a non-zero angular momentum, which is responsible for a misalignment of the rotation axis with respect to the initial magnetic field, and an effective turbulent diffusivity in the vicinity of the core. Both effects are responsible for a significant decrease in the magnetic braking, and they facilitate the formation of early massive disks. These disks can fragment even with μ ~ 5 at late times, in contrast to simulations of 1 M⊙ cores, where fragmentation is prevented for these values of μ. Slow asymmetric outflows are always launched, and they carry a mass comparable to that of the adiabatic first core. Conclusions. Because of turbulence-induced misalignment and magnetic diffusivity, massive disk formation is possible; nevertheless, their mass and size are much more reduced than for disks formed in unmagnetized collapsing cores. We find that for μ ≥ 5 fragmentation can occur.