When the Tweet expresses some abhorrent view (like "she deserves to be treated badly because she is $RACE", terrorism apologism, etc) and it garners hundreds of thousands of likes and retweets, it causes me quite a lot of stress, doubly so if it comes from or is endorsed by major publications or political officials.
Arguably, I shouldn't feel stress for things so far beyond my control, but at the same time I have a vested interest in my community and its future (and if so many people feel this way, it certainly _feels_ like a credible threat).
I think stress is natural reaction to learning that your community has abhorrent views, but you could train yourself to minimise that reaction in a stoic manner.
Training is in many ways a body thing as well as mind. E.g. learning to modulate fear and disgust reactions, heart rate, breathe so as to let the adrenalin spike pass (if you want), change muscle posture, etc.
And you should anyway on intellectual grounds, because words' intepretion is often different, at least in degree and shades and layers of meaning, than the intent of their author. For example "$X apologism" is often a misinterpretation of words someone was using to try to reduce the amount of $X in the world (due to the world and people being complicated). Many issues are "no matter what you say, or where you are coming from, someone will think you mean almost the opposite, and sometimes they have a vested interest in thinking that".
But reducing stress doesn't have to mean stop caring.
It can make you more effective at caring.
For anyone with an interest in their community and its future, I'd advocate for deciding which reactions to keep, maybe even develop (e.g. various ways to channel anger constructively, such as politics, community organising, creating new institutions), and training yourself to reduce stress and other intense reactions that you decide are not so helpful (e.g. anxiety from watching the news).
My main point is to encourage people to think of stress and anger as being a mix of constructive and destructive types.
My approach these days is to first try to overcome my anger and despair at knowing how many people seem to think what they do, and when I (eventually!) calm down, ask myself how can I affect the world to change this. I must admit, it is challenging these days because I've come to think of large numbers of my fellow humans as jerks who are happiest when damaging others!
Arguably, I shouldn't feel stress for things so far beyond my control, but at the same time I have a vested interest in my community and its future (and if so many people feel this way, it certainly _feels_ like a credible threat).