To give an example, there are a number of women working in the games space who are targeted every time they express any sort of view. Some of these threats are simply extraordinarily disgusting.
"Targeted"? What exactly do you mean by that?
If you mean that people disagree loudly and vigorously when they speak, well, welcome to being an adult.
If you mean that people threaten them, an actual, credible threat is a crime. And in such instance Twitter should be forwarding info to help the police to catch the criminal.
But hyperbolic speech -- even speech you or I may find "extraordinarily disgusting" -- is not a credible threat. If you don't want to read disgusting speech, Twitter lets you block people. We've had the solution for dealing with asshats on-line since the glory days of USENET. It sounds like this: plonk.