The growth and evolution of grains in the protostellar nebula is investigated within the context of turbulent-low mass disk models developed by previous investigators. Because of grain collisions promoted by the turbulent velocities, particles aggregate to millimeter size in times of order 1^3 yrs. During the growth the particles acquire a large inward radial velocity due to gas drag (Weidenschilling, 1977) and spiral into the sun. The calculations indicate that the final size of the particles does not exceed a few centimeters. This result is not very sensitive to the specific nebula parameters. For all conditions investigated it seems impossible to grow meter- or kilometer-sized bodies which could decouple from the gas motion. An additional argument is given which shows that only particles smaller than centimeter size can survive drift into the growing sun by being transported radially outwards by turbulent mixing. This agrees well with the maximum size of inclusions and chondrules. Since sedimentation of grains and subsequent dust disk instability is effectively inhibited by turbulent stirring, the formation of planetesimals and planets can not be explained in the above scenario without further assumptions.