Direct numerical simulation data obtained from two turbulent, lean hydrogen-air flames propagating in a box are analyzed to explore the influence of combustion-induced thermal expansion on turbulence in unburned gas. For this purpose, Helmholtz-Hodge decomposition is applied to the computed velocity fields. Subsequently, the second-order structure functions conditioned to unburned reactants are sampled from divergence-free solenoidal velocity field or irrotational potential velocity field, yielded by the decomposition. Results show that thermal expansion significantly affects the conditioned potential structure functions not only inside the mean flame brushes, but also upstream of them. Upstream of the flames, firstly, transverse structure functions for transverse potential velocities grow with distance r between sampling points more slowly when compared to the counterpart structure functions sampled from the entire or solenoidal velocity field. Secondly, the former growth rate depends substantially on the distance from the flame-brush leading edge, even at small r. Thirdly, potential root-mean-square (rms) velocities increase with decreasing distance from the flame-brush leading edge and are comparable with solenoidal rms velocities near the leading edge. Fourthly, although the conditioned axial and transverse potential rms velocities are always close to one another, thus, implying isotropy of the potential velocity field in unburned reactants; the potential structure functions exhibit a high degree of anisotropy. Fifthly, thermal expansion effects are substantial even for the solenoidal structure functions and even upstream of a highly turbulent flame. These findings call for development of advanced models of turbulence in flames, which allow for the discussed thermal expansion effects.