Sympathy is usually said when something sad happens. I.e.: when a friend's sister dies, you could say "you have my sympathy" to mean "I feel sorry for your loss/death in the family". I think saying "you have my support" is fine as is. If you can provide context there might be another phrase you can use but generally that's good as is.

"support" is not quite the same. "sympathy" is an emotion, a feeling. . The speaker is explaining that they also feel a loss, rather than some other feeling, such as being ambivalent or glad.