General Relativity reduces to Newtonian gravity as the curvature goes to zero, that is when you're very far away from objects relative to their masses, for slow non-relativistic objects like stars and galaxies.

Galaxies are typically so far away from another they're almost like point sources to each other, hence Newtonian gravity explains their motion very well.

However, inside galaxies things do not behave as expected, as stars in almost all the galaxies we've measured does not move like Newtonian (nor GR) behaves based on the matter in the galaxy we see. One alternative to the mainstream theories of dark matter is to modify Newtonian gravity, called MOND.

This work tested if MOND fit the motion of galaxies in galaxy clusters. They found it did not.

MOND already does not explain other phenomena that dark matter can so it's not terribly surprising. Here[1] is a nice accessible talk going through all the evidence for dark matter.

But it is technically a possibility that there's two things are going on, something MOND-like as well as dark matter, so worth checking.

[1]: https://pirsa.org/26030070

Special Relativity is an extension of Galilean/Newtonian mechanics (motion of projectiles and other objects) to the case where the object is travelling at speeds that are a fraction of the speed of light. It deals with non-accelerating frames of reference. Satelites need to use this to correct for time dilation effects, but tracking the trajectory of an arrow/etc. or a car/etc. travelling from one location to another then classical mechanics is sufficient.

General Relativity is an extension of Newtonian gravity. It is also an extension of Special Relativity to cover accelerating frames of reference. Satelites need to use this, as does tracking the orbit of Mercury. However, for the orbits of other planets and the moon, using Newtonian gravity is sufficient for a reasonable degree of accuracy, and is used for tracking things like equinoxes/solstices, full moons, etc..

At these scales (entire galaxies, very weak forces), it doesn't make a significant difference.

There are ways of adapting MOND to match general relativity, should it turn to be correct at explaining what it is supposed to explain (like the movement of galaxies).

General Relativity. It explained the anomaly in the precession of Mercury's perihelion, and the bending of starlight by the Sun (double the value predicted by Newton's law).

The test here is for the inverse square law of gravity. The rival theory in this case isn't GR, but MOND: https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics