  Fast radio bursts (FRBs) have been identified as extragalactic sources which
can make a probe of turbulence in the intergalactic medium (IGM) and their host
galaxies. To account for the observed millisecond pulses caused by scatter
broadening, we examine a variety of possible models of electron density
fluctuations in both the IGM and the host galaxy medium. We find that a
short-wave-dominated power-law spectrum of density, which may arise in highly
supersonic turbulence with pronounced local dense structures of
shock-compressed gas in the host interstellar medium (ISM), can produce the
required density enhancements at sufficiently small scales to interpret the
scattering timescale of FRBs. It implies that an FRB residing in a galaxy with
efficient star formation in action tends to have a broadened pulse. The scaling
of the scattering time with dispersion measure (DM) in the host galaxy varies
in different turbulence and scattering regimes. The host galaxy can be the
major origin of scatter broadening, but contribute to a small fraction of the
total DM. We also find that the sheet-like structure of density in the host ISM
associated with folded magnetic fields in a viscosity-dominated regime of
magnetohydrodynamic (MHD) turbulence cannot give rise to strong scattering.
Furthermore, valuable insights into the IGM turbulence concerning the detailed
spatial structure of density and magnetic field can be gained from the observed
scattering timescale of FRBs. Our results are in favor of the suppression of
micro-plasma instabilities and the validity of collisional-MHD description of
turbulence properties in the collisionless IGM.
