I've never heard a rational defense of moral relativism that made any sense. If there are no moral truths, then serial killers have done nothing wrong for example. If a moral relativist admits that there are some moral truths, then moral relativism is completely indefensible. At that point, you're just arguing over what is and what is not a moral truth.
Universal moral truths. Like absolutes. We can say killing is bad, but many would say killing a mass murderer currently on a murder spree would be more moral than letting them kill a bunch of people.
Just because there aren't moral truths doesn't mean a serial killer did nothing wrong. You seem to be stuck on finding a single contradiction and using that to dismiss everything else related as irrelevant. That's not actually how the world works.
Similarly in physics, the existence of non-newtonian fluids, doesn't invalidate Newton's work in fluid dynamics.
If there are no moral truths, then serial killers have done nothing wrong for example
This does not follow from moral relativism. Moral relativism simply states the morality of serial killers is determined by people rather than an absolute truth.
For example, if you add the detail of “serial killer of humans”, most societies would deem that morally wrong. In contrast, “serial killer of wasps” would be considered perfectly fine by many. A moral relativist would say the difference between these two is determined by society.
You can, of course, claim that murdering humans is not morally wrong. A moral absolutist might say “you’re wrong because X”, while a moral relativist might say “I don’t agree because X”.