I've read some articles which explain that we can't totally rely on Newtonian gravity concepts for understanding the motion of objects in space.
My question is how the use of relativity helps us to better explain observed motion of objects in space, and how it corrected Newtonian gravity results.
The orbital motion of Mercury is an obvious example, but the Wikipedia section Perihelion precession of Mercury is very advanced and difficult to understand.
Yes, the Mercury problem seems to be horribly complicated. The C19th Newtonian calculations, apparently predicted part of the perihelion precession, but not all of it.
C19th Newtonian gravity can be considered as being equivalent to a "curved-space" theory. If you use wavelengths of light as "rulers", they contract as the light passes into a more intense gravitational field due to gravitational blueshifting, so mapping the region with these "rulers" gives an apparent variable density of space, which can be used to predict gravitational lightbending as a refractive index effect due to the apparent variations in spatial density. Gravity-shifts were originally predicted by John Michell back in 1783, but were then apparently almost immediately forgotten about.
In 1911, Einstein rediscovered the gravity-shift idea, and realised that it unavoidably led to gravitational time dilation. This is now usually considered to be a GR effect, but Einstein presented the original argument as a Newtonian calculation for simplicity, some years before GR was finished (1915/1916), so it's arguably also a "Newtonian updated by Einstein" effect. (!)
In 1911, Einstein used the time dilation idea to say that light travelled slower in a more intense gravitational field, and used Huyghens' Principle to predict the same light-bending effect as before, but now due to variations in timeflow ("curved time" calculation).
However, this wasn't the whole story, because just because both calculations ("curved space" and "curved time") gave the same answer, this didn't immediately tell us whether they were the same effect calculated two different ways, or were two different effects that had to be multiplied together. With general relativity's curved spacetime, Einstein wanted the effects to be cumulative, roughly doubling the lightbending prediction, partly because this would also increase Mercury's precession, explaining the anomaly.
So GR ended up explaining the Mercury problem, and also successfully predicted the doubled gravitational light-bending effect, which was confirmed by Eddington shortly after (1919). Success!
However, doing comparative theory properly is still damned confusing, because ... do we define Newtonian theory as being the C19th version, or do we upgrade it with things like gravitational time dilation, that the theory really should have predicted in the C19th, but didn't until Einstein came along in the C20th? Some people would consider retrofitting the "good new bits" of GR onto NM to be cheating, and since an upgraded NM might be almost indistinguishable from GR, there's little obvious incentive to do it, because we don't need two different near-identical theories.
Einstein's general theory is in general definitely much better than C19th textbook Newtonian theory, partly because the C19th theory had to model light as particles ("ballistic emission theory"), in a way that didn't allow a sensible light-metric. With BET, light from differently moving objects would be thrown off at different speeds, and would continue moving at those different speeds indefinitely, so you had crazy things happening like different light-signals overtaking each other along the same path at the same time. Anarchy!
This had made people give up on Newtonian optics and focus more on aether theories in the late C19th, because these had a single orderly speed of light in any given direction in any location in space, as a function of the medium's properties. But unlike Newtonian theory, aether theories tended not to obey the principle of relativity (which was wrong, and bad). Lorentz then produced a relativistic aether theory in 1904, Einstein grabbed the math and deleted the aether aspect to produce special relativity in 1905, and then produced the general theory in ~1916, which kept the SR lightmetric but also had it curved by gravity, rotation and acceleration.
If we were to revive and update Newtonian theory, and turn it into a sibling of Einstein's general theory, we'd have to find some way to enforce wave-compatibility, and that'd make some things much more complicated than SR/GR. It may be possible but ... many GR people would say, why go to all that trouble to create something almost like GR, and more complex than GR, when we already have GR?
