Newtonian physics can predict fixed curves, you just have to assume that there is additional matter you don't see through a telescope, aka 'dark' matter. And as it turns out, if you then look at the cosmic microwave background, you see that cosmology works a lot better if you have an additional uncharged field, that has a similar mass density as the missing matter in galaxy. When you then look at the large scale structure of galaxies, then it turns out that it looks like there is an additional component to galaxy formation, that again seems to be uncharged but otherwise looks a lot like normal matter.

Now, the picture is only mostly as neat now than it was a few years ago, since there is now quite a bit strain with particle physics, but still everywhere in astrophysics it looks like there is some component missing. If that component is dark-matter or a modification of gravity or a remnant of something else is a question about which story journalists want to tell. The thing is, nature does not have to adhere to how we name the left or right hand side of an equation, and that is on a phenomenological level really the difference between mond and dark matter, you either write the term on the gravity side or on the matter side of Newton's law.

It is not that physicists invent an invisible component to make a model work, physicists look at the universe and find hints of a missing matter component everywhere. It is just that journalists love to claim that scientists are puzzled.