We present the results of a numerical investigation of three-dimensional homogeneous and isotropic turbulence, stirred by a random forcing with a power-law spectrum, Ef(k)~k3-y. Numerical simulations are performed at different resolutions up to 5123. We show that at varying the spectrum slope y, small-scale turbulent fluctuations change from a forcing independent to a forcing dominated statistics. We argue that the critical value separating the two behaviors, in three dimensions, is yc=4. When the statistics is forcing dominated, for yyc, we find the same anomalous scaling measured in flows forced only at large scales. We connect these results with the issue of universality in turbulent flows.