We investigate the feasibility of detecting and probing various components of the ionized intergalactic medium (IGM) and their turbulent properties at radio frequencies through observations of scatter broadening of compact sources. There is a strong case for conducting targeted observations to resolve scatter broadening (where the angular size scales as $sim u^{-2}$) of compact background sources intersected by foreground galaxy haloes and rich clusters of galaxies to probe the turbulence of the ionized gas in these objects, particularly using Space VLBI with baselines of 350,000 km at frequencies below 800 MHz. The sensitivity of the Square Kilometre Array (SKA) allows multifrequency surveys of interstellar scintillation (ISS) of $sim 100 ,mu$Jy sources to detect or place very strong constraints on IGM scatter broadening down to $sim 1, mu$as scales at 5 GHz. Scatter broadening in the warm-hot component of the IGM with typical overdensities of $sim 30$ cannot be detected, even with Space VLBI or ISS, and even if the outer scales of turbulence have an unlikely low value of $sim 1$ kpc. Nonetheless, intergalatic scatter broadening can be of order $sim 100, mu$as at 1 GHz and $sim 3, mu$as at 5 GHz for outer scales $sim 1$ kpc, assuming a sufficiently high source redshift that most sight-lines intersect within a virial radius of at least one galaxy halo ($z gtrsim 0.5$ and $z gtrsim 1.4$ for $10^{10} {rm M}_odot$ and $10^{11} {rm M}_odot$ systems, following McQuinn (2014)). Both Space VLBI and multiwavelength ISS observations with the SKA can easily test such a scenario, or place strong constraints on the outer scale of the turbulence in such regions.
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