The ability to probe quantum systems on short timescales is central to the advancement of quantum technology. Here we show that this is possible using an off-resonant dispersive probe. By applying a magnetic field to an atomic vapour the spectra of the group index for left and right circularly polarized light become displaced, leading to a slow-light Faraday effect that results in large dispersion and high transmission over tens of gigahertz. This large frequency range opens up the possibility of probing dynamics on a nanosecond timescale. In addition, we show that the group index enhances the spectral sensitivity of the polarization rotation, giving large rotations of up to 15π rad for continuous-wave light. Finally, we demonstrate dynamic broadband pulse switching by rotating a linearly polarized nanosecond pulse by π/2 rad with negligible distortion and transmission close to unity.