Newton's laws can explain Mercury's orbit over the course of just one orbit, just fine. It's the perturbations that it can't explain well over the course of multiple orbits, which end up making Mercury's orbit precess around from orbit to orbit. There is a sideways component of the orbital acceleration that Newton can't explain. As far as Newton is concerned, gravity should only have lines of force radially, not tangentially. That works perfectly if the speed of light (and gravity) were infinite, but with a limited speed of light, it takes time for gravity to reach between objects, and by the time the gravitational force reaches the object, it starts coming at it from non-radial-only direction. The force of gravity is sort of bent in from a different direction.