We present a comparative experimental study of a turbulent flow developing in clear water and dilute polymer solutions (25 and 50 wppm polyethylene oxide). The flow is forced by a planar grid that oscillates vertically in a square container of initially still fluid. The two-component velocity fields are measured in a vertical plane passing through the center of the tank by using time resolved Particle Image Velocimetry (PIV). We obtain a lower entrainment rate for polymer solutions as compared to clear water. Extending arguments based on similarity and fractal theory to the case of dilute polymer solutions, we derive a relation between the entrainment rate and the fraction of input energy dissipated by the polymers.