Even small masses, like the earth and your body, warp space-time, but the effect is small. That's why we don't experience light doing funny things in our every day life.
The space-time warp that earth causes is so small that light beams appear to go in straight lines. In other words, where you point a flashlight is where the light goes. And the light from far away stars arrives in our sky as steady pin-pricks of light. When you take a plane trip or even go to the moon the stars do not appear to shift around (as distant trees do when you drive in your car) because the stars are incredibly far away and because space-time in the solar system is fairly flat.