In this paper, we propose a novel effective optical smoothing scheme to suppress laser plasma instabilities (LPIs) by time-dependent polarization rotation (TPR) on a picosecond timescale. The polarization rotation with time-dependent frequency is generated by the superposition of chirped light pulses with dynamic frequency shift and counter-rotating circular polarization. Compared to light without polarization rotation or pulse chirping, such superposed light with TPR has a broader spectrum and lower temporal coherence. Using the one-dimensional fluid laser-plasma-instability code (FLAME) and PIC simulation, TPR is demonstrated working well in suppressing parametric backscattering, which provides an effective approach to suppress LPIs. In the meantime, a significant improvement of irradiation uniformity of the chirped pulses is achieved by the introduction of proper spatial phase modulation and grating dispersion.