The description of suffering that I identify with is a desire for things to be other than they are. I may experience physical pain and want it to end; that's a type of suffering. I may desire for my annoying co-worker to stop bragging about his vacation and get to the point of the meeting; that's another kind of suffering. I know that there are people in concentration camps in the US, right now, suffering who knows what all torments and indignities of their own that I wish to end; that gives me some suffering even though I am not there with them, nor do I know their situation exactly.
All of these things, and millions more, are things about my experience of the world that I desire to be different. The fact that they aren't as I would like is upsetting, and I must either change them or, if I can't, I must suffer them.
I can't think about them all the time. It used to be that I thought my only options were to think about them all the time, which is exhausting, or not think about them at all, which feels callous to the point of psychopathy. A large part of my mental health journey has been tuning how I identify, deal with, and compartmentalize the suffering I do, on my own behalf and on the behalf of others.
I don't think suffering is bad, or to be avoided. It is a thing to become familiar with, like anger (itself usually a response to suffering), that needs to be identified for what it is and channeled. Suffering is very like an emotion in that it is an indicator in the brain that you want something different. Sometimes, identifying the suffering makes it easier to identify and remedy the cause. Other times, the cause is so huge and overwhelming (the whole US situation right now), that I just have to deal with suffering the whole thing while I do what small things I can to make things a little bit better.