New measurements of the beam normal single spin asymmetry in the electron elastic and quasielastic scattering on the proton and deuteron, respectively, at large backward angles and at ⟨Q^{2}⟩=0.22 (GeV/c)^{2} and ⟨Q^{2}⟩=0.35 ( GeV/c)^{2} are reported. The experimentally observed asymmetries are compared with the theoretical calculation of Pasquini and Vanderhaeghen [Phys. Rev. C 70, 045206 (2004).PRVCAN0556-281310.1103/PhysRevC.70.045206]. The agreement of the measurements with the theoretical calculations shows a dominance of the inelastic intermediate excited states of the nucleon, πN and the Δ resonance. The measurements explore a new, important parameter region of the exchanged virtual photon virtualities.