"the curvature of spacetime is caused by the accelerating movement of dark energy"

No. This is not correct.

Or is it that we do but the effect is very small?

Yes, the effect of dark energy is too small to influence ordinary physics around us.

"the curvature of spacetime is caused by the accelerating movement of dark energy"

No. This is not correct.

Or is it that we do but the effect is very small?

Yes, the effect of dark energy is too small to influence ordinary physics around us. You can measure its effect by using very powerful telescopes to see galaxys beyond our galaxy.