The orbital speed depends on the mass(M) you are orbiting and your distance from the center of that mass. Increasing M has the effect of increasing the orbital speed and increasing the distance has the effect of decreasing it If you are in a spherical body, or a disc, the only mass contributing to M is that part of the body closer to the center than you are. With a spherical body the mass closer to the center increases by the cube of your distance from the center, so when you move away from the center, the effect of increasing M exceeds the effect of increasing the distance, so orbital speed goes up until you are outside the body. Then M remains constant even while the distance increases and orbital speed starts going down. With a disk, M does not go up as fast with r, and in fact, the two effects would cancel and you would expect the orbital speed to stay constant with distance. A spiral galaxy is neither purely spherical or disc shaped. It is a disc with a spherical bulge in the center. Inside the bulge, the effect on orbital speed in what you would expect to find in a sphere. Once you get out of the bulge, you have to consider both the effects the mass of the bulge has and the increasing mass added by the disk as you move outward. The effect of moving away from the mass of the bulge has the effect of decreasing the orbital speed, The addition of mass due to the disc will lessen this effect but will not be enough to counter it completely. Thus you should see the orbital speed fall off with distance as you increase distance from the bulge. Not as fast as it would if all the mass were concentrated in the bulge, but still falling off.

The whole point is is that when you take into account all the mass of the visible galaxy and the way it is distributed, you should get a certain pattern to the way orbital speed changes with distance. The fact that the actual orbital speeds we measure do not follow this pattern indicates that there is mass involved that we do not see. And it is just not a matter of more mass. If it were just a matter of getting the scaling factor of the mass wrong, you would get different values for the orbital speeds, but the pattern would stay the same. It is the fact that the pattern itself differs that indicates that the extra mass does not follow the same distribution as the matter we see in the galaxy.