How do I tell someone who has gone through a loss that I'm sorry and I care without offending them?

I have a confession to make again: I’m extremely fearful of situation when I have to comfort someone who has just gone through a loss. It’s not because I don’t care for them. I care, A LOT, but I’m just so scared of saying the wrong thing and offending them. When someone comes toward me with a story of their loss, I literally feel my brain freeze. I don’t know what to say to comfort them. I fear that whatever I say will end up hurting them even more. The only thing I can say is “I’m sorry for your loss”, but I’m scared that is not enough and I am supposed to say something more, but I don’t know what.

As a result, I try to avoid situations like that. I feel horrible for not being able to comfort, but I’m just afraid that I will just end up making the situation worse.

Is there any way for me out of this dilemma?