I haven't really thought of this as much of a problem, but I'm wondering if you are confusing simple compassion with trying to be a support person for your therapist.

I've had therapist lose members of their family. I've had them go through an illness, etc. I showed compassion to them like I would anyone I know on that level: I simply said "sorry for your loss" or "I hope you get to feeling better." I didn't expect them to tell me details, nor would that have been appropriate. And I knew I wasn't going to be their support person; they have other family members, closer friends, etc. who they lean on for that kind of support. They thanked me for the well wishes or condolences, and we moved on. That really isn't any different than I would approach most people. Unless you are a very close friend or family member, it isn't my place to do much more than say a few words and allow the other person to use their more personal support system for whatever talking or needs they have.