ABSTRACT The pulse morphology of fast radio bursts (FRBs) provides key information in both understanding progenitor physics and the plasma medium through which the burst propagates. We present a study of the profiles of 33 bright FRBs detected by the Australian Square Kilometre Array Pathfinder. We identify seven FRBs with measureable intrinsic pulse widths, including two FRBs that have been seen to repeat. In our modest sample, we see no evidence for bimodality in the pulse width distribution. We also identify five FRBs with evidence of millisecond time-scale pulse broadening caused by scattering in inhomogeneous plasma. We find no evidence for a relationship between pulse broadening and extragalactic dispersion measure. The scattering could be either caused by extreme turbulence in the host galaxy or chance propagation through foreground galaxies. With future high time resolution observations and detailed study of host galaxy properties, we may be able to probe line-of-sight turbulence on gigaparsec scales.