As I understand things (but don't just take my word for anything), as you measure the speed of material (e.g. stars) rotating round, you do not find what you would expect. Ignoring the material close to the centre, as you move out you would expect the speed to decline (i.e. further out, slower speed) - as you said. However, this is found not to be the case (talking about Spiral Galaxies here). What is found is that as you move out so everything is travelling at around the same speed. Talking about distance per unit time speed here not angular speed. So why is material further out not travelling slower - and currently many think it can only be explained by dark matter throughout the galaxy and extending far out beyond the radiating regions. Of course there are other theories (e.g. MOND) and I cannot argue the merits of different explanations.
I just mentioned it in relation to the reasons for dark matter existing. The galactic rotation is interesting as it also throws light of the spiral arms. Because, with the speeds being moderately constant as one moves out from the centre, then why to the spiral arms not wind tighter and tighter (as material nearer the centre is taking a lot less time to go round than material further out) - but then one is into different discussions (off-topic).
I personally don't see dark matter is a massive issue (not a matter of "belief"

). Maybe some think of it as just some kind of really exotic stuff that needs all sorts of weird complex physics to explain it. I just think of it as matter that does not radiate (or interact with radiation) - i.e. that we cannot directly see. But just because we cannot see it does not mean we cannot detect it indirectly (e.g. through its gravitational effects). Maybe it does involve all sorts of complex physics as well but I can see no problem with thinking there is matter around we cannot directly see but can detect. (Using the term "see" in the broadest sense in terms of detecting through radiation).
Just google "galactic rotation curves".
Ian
(Still not been reprimanded)