Dark matter is measurable, and the reason we know it exists is because of measurements in observational astronomy. When astronomers began measuring the motion of other galaxies they found that there was a drastic difference between the luminous mass of a galaxy and its gravitating mass. This is why originally dark matter was referred to as "missing mass". Dark matter does create a gravitation lensing effect (that's how it affects light) but it also interacts gravitationally with matter around it, and these effects are all measurable.

There are many ways to approach the reality of dark matter, and I personally reject these types of "God of the Gaps" approaches in favor of theories derived through physical and mathematical rigor (though there's no way to know what approach will prevail in the end). For example, one interpretation of string theory predicts that the graviton (the as-yet unobserved force-carrying particle for the gravitational force) is unique in that it is multi-dimensional (able to traverse more than the 3 spacial dimensions we observe and exist in the other dimensions postulated by string theory) and that dark matter is a gravitational "shadow" (for lack of a better term) of mass contained in other dimensions.