We present a detailed experimental and theoretical study on the relativistic nondipole effects in strong-field atomic ionization by near-infrared linearly polarized few-cycle laser pulses in the intensity range of 10^{14}-10^{15} W/cm^{2}. We record high-resolution photoelectron momentum distributions of argon using a reaction microscope and compare our measurements with a truly ab initio fully relativistic 3D model based on the time-dependent Dirac equation. We observe counterintuitive peak shifts of the transverse electron momentum distribution in the direction opposite to that of laser propagation as a function of laser intensity and demonstrate an excellent agreement between the experimental results and theoretical predictions.