At low temperatures the dynamical degrees of freedom in amorphous solids are tunneling two-level systems (TLSs). Concentrating on these degrees of freedom, and taking into account disorder and TLS-TLS interactions, we obtain a “TLS glass,” described by the random-field Ising model with random 1/r^3 interactions. In this paper we perform a self-consistent mean-field calculation, previously used to study the electron-glass (EG) model [A. Amir et al., Phys. Rev. B 77, 165207 (2008)]. Similarly to the electron glass, we find a 1/λ distribution of relaxation rates λ, leading to logarithmic slow relaxation. However, with increased interactions the EG model shows slower dynamics whereas the TLS-glass model shows faster dynamics. This suggests that given system-specific properties, glass dynamics can be slowed down or sped up by the interactions.