Yup [1]
> My understanding is that based on observations of what can be seen galaxies move as if there's move stuff in them??
Yup [2]
> If you hypothesis that this extra stuff is dark matter but can't detect it, how do you know it's real?
It produces certain effects in various different enviromennts or domains, which lends more support to that hypothesis. By certain different domains, I mean like Galaxy and cluster dynamics, gravitational lensing around galaxies and clusters and the precise fluctuations seen in the cosmic microwave background, these are all very different regimes but all seem to support the hypothesis of extra mass which is not interacting electromagnetically.
> How have you ruled out all the normal stuff that doesn't give off light or stuff we can't see at the distances we're looking over?
There are research efforts to look for exactly this kind of stuff; Massive compact halo object (MACHOS). However they don't see enough of it to account for the observations we see in galaxies and galaxy clusters. [3]
[1] https://lss.fnal.gov/archive/2022/conf/fermilab-conf-22-180-... [2] https://en.wikipedia.org/wiki/Dark_matter#Observational_evid... [3] https://en.wikipedia.org/wiki/Massive_compact_halo_object#De...