We study incompressible imbalanced magnetohydrodynamic turbulence in the presence of background velocity shears. Using scaling arguments, we show that the turbulent cascade is significantly accelerated when the background velocity shear is stronger than the velocity shears in the subdominant Alfvén waves at the injection scale. The spectral transport is then controlled by the background shear rather than the turbulent shears and the Tchen spectrum with spectral index −1 is formed. This spectrum extends from the injection scale to the scale of the spectral break where the subdominant wave shear becomes equal to the background shear. The estimated spectral breaks and power spectra are in good agreement with those observed in the fast solar wind. The proposed mechanism can contribute to enhanced turbulent cascades and modified −1 spectra observed in the fast solar wind with strong velocity shears. This mechanism can also operate in many other astrophysical environments where turbulence develops on top of non-uniform plasma flows. © 2016, Springer Science+Business Media Dordrecht.