We have studied turbulent drag reduction using the technique of photon-correlation homodyne spectroscopy to measure turbulent velocity gradients in a dilute polymer solution. A large anisotropic suppression of turbulent velocity differences is found in the bulk region of the turbulent fluid and along the direction of the mean flow. The suppression effect is associated with an enhancement of the turbulent drag reduction. This anisotropy caused by polymer molecules in turbulent flow may be important in understanding turbulent drag reduction.