The renormalization group (RNG) method is used to study the physics of two‐dimensional (2‐D) magnetohydrodynamic (MHD) turbulence. It is shown that, for a turbulent magnetofluid in two dimensions, no RNG transformation fixed point exists on account of the coexistence of energy transfer to small scales and mean‐square magnetic flux transfer to large scales. The absence of a fixed point renders the RNG method incapable of describing the 2‐D MHD system. A similar conclusion is reached for 2‐D hydrodynamics, where enstrophy flows to small scales and energy to large scales. These analyses suggest that the applicability of the RNG method to turbulent systems is intrinsically limited, especially in the case of systems with dual‐direction transfer.