It’s not only one prediction that does not fit to observations, it’s many different ones. For dark matter it’s that galaxies rotate too fast, galaxies in clusters move too fast, gravitational lenses bend light too strongly, and neither the cosmic microwave background nor galactic filaments would look like we observe them without dark matter. I explained this in detail in an earlier video.
For dark energy the shooting gun signature is that the expansion of the universe is getting faster, which you can find out by observing how fast supernova in other galaxies speed away from us. The evidence for dark energy is not quite as solid as for dark matter. I explained this too in an earlier video.
So, what’s the scientist to do when they are faced with such a discrepancy between theory and observation? They look for new regularities in the observation and try to find a simple way to explain them. And that’s what dark energy and dark matter are.Sabine Hossenfelder, “Are dark energy and dark matter scientific?” at BackRe(Action)
Ultimately, of course, they may seem “scientific” principally because they are ideas that appear in a certain form. The question of whether the terms “dark matter” and “dark energy” correspond to anything that actually exists could be a different one.