Cryogenically cooled buffer gas beam sources of the molecule thorium monoxide (ThO) are optimized and characterized. Both helium and neon buffer gas sources are shown to produce ThO beams with high flux, low divergence, low forward velocity, and cold internal temperature for a variety of stagnation densities and nozzle diameters. The beam operates with a buffer gas stagnation density of ∼10(15)-10(16) cm(-3) (Reynolds number ∼1-100), resulting in expansion cooling of the internal temperature of the ThO to as low as 2 K. For the neon (helium) based source, this represents cooling by a factor of about 10 (2) from the initial nozzle temperature of about 20 K (4 K). These sources deliver ∼10(11) ThO molecules in a single quantum state within a 1-3 ms long pulse at 10 Hz repetition rate. Under conditions optimized for a future precision spectroscopy application [A. C. Vutha et al., J. Phys. B: At., Mol. Opt. Phys., 2010, 43, 074007], the neon-based beam has the following characteristics: forward velocity of 170 m s(-1), internal temperature of 3.4 K, and brightness of 3 × 10(11) ground state molecules per steradian per pulse. Compared to typical supersonic sources, the relatively low stagnation density of this source and the fact that the cooling mechanism relies only on collisions with an inert buffer gas make it widely applicable to many atomic and molecular species, including those which are chemically reactive, such as ThO.