Which experiments confirm that the Born rule is $|\psi|^2$ rather than $|\psi|$? It seems like some experiments on quantum systems, like the electron $g-2$ measurement, do not rely directly on the Born rule, since they are more so measuring inherent characteristics of the evolution of the wavefunction.
Whereas, an experiment that obviously does rely on the Born rule is the single-electron double-slit experiment, since in principle you could tell whether the distribution of electrons looks more like $|\psi|^2$ versus, let's say, just $|\psi|$.  However, I've watched a couple videos of that experiment, and to me it doesn't look precise enough to distinguish between $|\psi|^2$ and $|\psi|$.  (I mean, it's a mammoth achievement just to get that experiment working at all)
So maybe it's just my utter lack of experimental knowledge, but I'm wondering, can we really be sure the probability goes as $|\psi|^2$ and not $|\psi|$?
 A: I might be missing the depth somewhere, but I would suggest the electron double-slit experiment first: $|\sin(kx)|$ clearly looks different as compared to $\sin^2(kx)$.
The second example is the Malus' law in optics. Polarizers are performing the quantum measurement on photons' polarization state. The fact that the measured intensity goes as $\cos^2(\theta)$ seems like a proof enough to me of the "squared version" of the Born rule.
A: Any photon-counting diffractive spectroscope. We see a rate of photon arrival proportional to the intensity of the diffracted wave, that is the square of the amplitude.
A: 
So maybe it's just my utter lack of experimental knowledge, but...

It's effectively unrelated to your knowledge (or lack thereof) of experiments.

...I'm wondering, can we really be sure the probability goes as $|\psi|^2$ and not $|\psi|$?

By definition, given a wavefunction solution to the Schrodinger equation $\psi(x_1, x_2, \ldots)$, the probability density is (see, e.g.,  Messiah, Quantum Mechanics, Volume 1, Chapter IV, Section 2):
$$
|\psi(x_1, x_2, \ldots)|^2
$$
You could re-write a bunch of assumptions and conventions and thereby replace $|\psi|^2$ with some other monotonic function of $|\psi|^2$ and call that a probability density instead.
But, all the the conventions we use in practice are already directed at using $|\psi|^2$ as the probability density, not some other function.
