I was aware that there were observations of galaxies that required coming up with something to reconcile them with the known laws of physics. I still have a question in my mind about whether dark matter is the best solution to that problem. Since then, has the dark matter hypothesis been able to make predictions that have been experimentally or observationally verified?
Yup. Quite a lot, actually.
Another bit of evidence comes from X-ray observations of large clusters of galaxies. They tend to collect hot gas in the middle, and that can be used to measure their mass, directly from the X-ray temperature.
It's a lot more mass than the visible matter can account for.
More recently, the mean density of the universe and separately the "baryonic" (normal matter) density can be measured from fluctuations in the microwave background. The shape of the spectrum is not consistent with a universe where all the matter interacts electromagnetically (and all normal matter does). People made predictions about the shape of the spectrum 20 years ago, but WMAP only recently measured it at small enough scales to tell whether the earlier dark matter measurements from big bang nucleosynthesis were correct (both measurements are sensitive in different ways to the ratio of normal to total matter, and give rather precise and consistent results).
Note that not all normal matter is luminous. The CMB results tell you about
all normal matter, even the stuff you can't see in emission, by its effect in absorption. If it doesn't absorb or emit, it's dark matter; that's what dark matter means.
There is more. I'll steer you to a recent popular book for an exhaustive list. Lawrence Krauss' "A Universe From Nothing" appears to be fairly up to date.
You can go to Brian Greene's book if you want to get really esoteric (and keep in mind that not everyone buys into string theory).
