If gravity is the curvature of spacetime it should bend everything equally. To clarify my point I would like you to imagine two scenarios. Think of a bird flying in the storm while the wind is blowing sideways. Because the wind is an active force and the bird is fighting against it, the amount by which it's trajectory is bent should depend on it's momentum for a given strength of wind. Qualitatively speaking, more the momentum the less it bends. Now think of a train moving on a curved track. It should always bend by a fixed degree regardless of it's velocity or mass because the path itself is bent by that precise degree.
I am aware that in my examples I have only considered curvature in SPACE but the theory of relativity talks about curvature in SPACETIME and I am guessing that every object's trajectory through spacetime is indeed equally bent due to gravity. Can someone explain?
EDIT: After reading some comments, I have realized that perhaps my choice of words wasn't that great in the original question. I know that different object will follow different geodesic trajectories based on their initial conditions. What I was interested in is the CHANGE in trajectories due to spacetime curvature. The term I should have used in the question is acceleration. For example, the gravitational acceleration an object will experience near the surface of the earth is g regardless of it's mass or initial velocity. So it's natural to expect that every object will experience the same acceleration when they are moving sideways with respect to a gravitational body. In other words the "bending" should be equal. Hope it's a little clearer now.