I may be completely wrong here, but is your ex doing a fair bit of stirring here?
"both her and my ex think I should leave my partner"
"My ex is all the time telling my daughter I don't care about her"
You have a fairly normal, bright teenager on your hands who even without the stirring from your ex is going through the difficult years and instead of supporting (her as well as you) your ex is using this to cause trouble.
Your current partner sounds as though he is coming on way too strong considering his position in the family. (Not her father, not your husband). How dare he talk like that about your daughter (she behaves like a dog).
Tell both men to back off, they are making an already volatile situation worse.
Both of them are 'peeing on lamposts', .trying to be the alpha male to the detriment of the long term happiness of you and your daughter.
Right now you are the person who knows your daughter best. Follow your own instincts in dealing with this.