Why would GR's prediction for galaxy rotation curves differ from Newtonian gravity's in spite of prevailing intuition from dimensional analysis? Until recently, I had assumed that the predicted (non-dark matter) galaxy rotation curves shown in plots such the one on the Wikipedia article were calculated with general relativity. However, I now understand that Newtonian gravity is used, the argument being that the relativistic corrections should be small (e.g. here and here).
However, I then read the last few paragraphs of the "Alternatives to dark matter" section of the Wikipedia article and found that, at least in a few cases, studies such as this, this and this which have gone beyond the dimensional analysis argument and actually pursued GR's predictions in more detail have found evidence that GR itself might reduce the need to invoke the existence of dark matter to the extent that is often done. It surprised me that I hadn't heard about such findings before (in fairness, a couple of them are quite new).
If these results are correct, why would the prediction from GR not be well approximated by Newtonian gravity despite the dimensional analysis argument, which is often used to approximate the relative magnitude of relativistic corrections, suggesting that it should? It does concern me that similar arguments made all over physics could be crude/erroneous in some non-trivial way.
 A: I read Cooperstock & Tieu (2006) and will discuss it a bit.
Before getting started, we should remember that galactic rotation curves are not the only evidence for dark matter.  Gravitational lensing surveys and measurements of the cosmic microwave background both point to the existence of dark matter.  And all three (rot. curves, lensing, CMB) agree on the approximate ratio of baryonic matter to dark matter.  When a single model can explain multiple phenomena, that is strong evidence in favor of the model.
Background
The Newtonian gravity is a good model for motion around a point mass in the weak field, slow speed limit: $\frac{GM}{r}\sim v^2\ll 1$ (in units where $c=1$).  Starting from the Schwarzschild metric, one can take the appropriate limit and get Newtonian gravity.
Newtonian gravity is a linear theory, so the gravitational potential due to multiple point masses is the same as the sum of the individual gravitational fields.  This is because the field equation is a linear differential equation:
$$ \nabla^2 \varphi = 4\pi G \rho,$$
where $\varphi$ is the potential and $\rho$ is the mass density.
The usual method to determine the rotation curve of a galaxy with Newtonian gravity relies on a few assumptions: the gravitational field is weak, the galactic material is moving slowly.  There's also a hidden assumption that non-linear effects from GR are small.  This is true for 2-body dynamics, where you can track the appearance of non-linear terms in the post-Newtonian expansion.
A relativistic galactic model
Instead of modeling the galaxy as a collection of Newtonian point masses all orbiting their collective center of mass, Cooperstock & Tieu want to start with a fully relativistic, rotating cloud of dust.  This is related to the van Stockum dust metric.  Then they can take an appropriate limit of the relativistic system to get rotation curves.  If the Newtonian assumption was right all along, then their method should give the same result as the Newtonian one.  But they get something different.
The metric for their rotating dust contains non-zero, off-diagonal terms in $g_{t\phi}$.  This is the same type of term that results in frame dragging in the Kerr metric.  They define the important part of that off-diagonal term as $N$.  If all of the terms containing $N$'s were suppressed by one of the small factors, $\frac{GM}{r}$ or $v^2$, then none of this would matter.  To first order, Newtonian gravity would be fine.  Cooperstock & Tieu say that that is not the case.  The off-diagonal term $N$ matters.
The contribution of $N$ is non-linear.  You can't just add up the gravitational contribution from each dust particle to get the answer using Newtonian gravity.
Did they make a mistake?
The second half of the paper attempts to address concerns from other scholars.  There are a lot of concerns, and much of them are quite technical.  Without digging into the whole thing in more detail I can't tell for sure who is right.
The most convincing criticism to me is that the authors didn't start from the true, fully relativistic metric, but jumped ahead to an already approximate, weak field state.  Bratek et al (2006) claim that there is no Newtonian limit to a fully relativistic stationary, axisymmetric, and asymptotically flat spacetime of dust.  So the whole endeavor is flawed from the start.
Another common criticism is that their solution has a discontinuity in $\frac{\partial N}{\partial z}$ at $z=0$.  This leads to a singular surface, basically a thin sheet of mass, rendering the whole thing unphysical.  The mass distribution isn't really just dust, there's some exotic matter in the $z=0$ plane.  Cooperstock & Tieu argue this is a mathematical artifact, not a real problem.
One way to fix this singular surface is to force $\frac{\partial N}{\partial z} = 0$ at $z=0$, but this results in a loss of asymptotic flatness.  This points me right back to the Bratek paper...
A: The Cooperstock-Tieu paper doesn't really justify this in any detail. They show that the metric is proportional to $G^{1/2}$, where $G$ is the gravitational constant, so they say that this makes it a nonlinear problem (which is true in this sense), and also that this makes the self-gravitating type of problem different from a case like the solar system (which is also true). They then make a massive non sequitur in claiming that this somehow invalidates Newtonian gravity for this purpose. It's true that Newtonian gravity is a linear level of approximation to GR in the sense that the Newtonian field equations are linear. But that doesn't mean that Newtonian gravity somehow breaks down in self-gravitating systems where the potential of a self-consistent solution is a nonlinear function of the mass density.
There is a rebuttal paper by Rowland, "On claims that general relativity differs from Newtonian physics for self-gravitating dusts in the low velocity, weak field limit". It doesn't seem to be on arxiv, but you can find it on researchgate.net through a google scholar search.
In general, if you come across a "gee whiz" claim like this that doesn't seem to make sense, you can check it by doing a google scholar search on the paper, look for papers that cite it, and see if they contradict it.
Note that galactic rotation curves are not even our only evidence of dark matter. We also have cosmological evidence from big bang nucleosynthesis and CMB fluctuations, as well as other observations of galactic dynamics that aren't rotation curves.
A: in fact, if we apply gravitomagnetism (a universally accepted weak field, low velocity expansion of GR) to realistic (i.e, observationally motivated) stellar disk models, the GR corrections to the rotation curves are of the order of 10^-6, as expected. The mathematical treatment can be found in arxiv 2207.09736 "On the rotation curve of disk galaxies in GR". Physically, disk galaxies are in a very weak, low velocity state, with v/c ~ 0.001. Suggesting that GR can reproduce a flat rotation curve at large radii implies an hypothetical effect of (say) several 100%, that should be produced by a (v/c)^2 ~ 10^-6 perturbation term. The definite proof of such an effect would be formidable from several points of view, arguably the less important of all being the absence of DM. The claim would be a proof that 1) physically, newtonian gravity is not the limit of GR in the weak field, low velocity regime (a cornestone of GR) 2) mathematically, the field equation of GR would describe a singular perturbation problem (similar to the behavior of Navier-Stokes equations in the limit of zero viscosity, when compared to the perfectly inviscid fluidodynamical equations.
In addition, there are indications of the existence of DM based on GR, in astronomical systems with low rotation (gravitational lenses in velocity dispersion supported elliptical galaxies and clusters of galaxies), in excellent agreement with the newtonian predictions based on hydrostatic equilibrium of the hot, X-ray emitting gas atmospheres of these systems.
As a final historical remark (remark, not a physical/mathematical proof!) it is curious to notice that the first indications of the existence of DM in astronomical systems can be traced back to 1933 (if we exclude some earlier prescient statement due to Poincare himself), when Albert Einstein was alive. These means that the claimed, surprising effect of GR remained unnoticed not only to Einstein, but also to scientists in the league of Fermi, Chandrasekhar, Landau, Feynman, only to mention a few of them.
A: You state:

GR itself might reduce the need to invoke the existence of dark matter to the extent that is often done

. . .

have gone beyond the dimensional analysis argument and actually pursued GR's predictions in more detail have found evidence that GR itself might reduce the need to invoke the existence of dark matter to the extent that is often done.

italics mine
The theoretical formulations of physics, in this case Newtonian realm to General relativity realm, can be shown to be consistent, and Occam's razor is used by the modelers. More complicated models could always be found, and it would be difficult to refute them, as astrophysics is not a laboratory experiment.
To reduce the need of invoking dark matter, does not eliminate the need, i.e. does not show that the rotation curves can be fitted with general relativity. If it did eliminate the need, Ocam's razor would choose this theory for the mainstream formulation of the rotation curves.

if these results are correct, why would the prediction from GR not be well approximated by Newtonian gravity despite the dimensional analysis argument, which is often used to approximate the relative magnitude of relativistic corrections, suggesting that it should.

Other predictions on Newtonian gravity are not approximated by GR, in the sense that GR is much more accurate for the specific effect. That is how we have the GPS system working . It all depends on the specific values of the specific problem.

It does concern me that similar arguments made all over physics could be crude/erroneous in some non-trivial way.

If you look at  this example of a history of physics, you will see that slowly measurements and observations have eliminated a lot of crude approximations, beginning with the atoms of Democritus. So one expects this process to be ongoing, getting better and better fits to data and observations with the theories used. It all depends on orders of magnitude and errors both for predictions and data.
An interesting point on the mathematical description of rotational curves is made in this   paper:

We consider the application of quantum corrections computed using renormalization group arguments in the astrophysical domain and show that, for the most natural interpretation of the renormalization group scale parameter, a gravitational coupling parameter G varying $10^{−7}$ of its value across a galaxy (which is roughly a variation of $10^{−12}# per light-year) is sufficient to generate galaxy rotation curves in agreement with the observations.

Gravity has not been definitely quantized, and quantization is a realm that is open for investigation and the paper may be relevant once it is .
So it might be, once quantization of gravity stops being an effective theory, a model as the one above might eliminate the need to look for particles to model the galaxy rotation curves.
We have to wait and see.
