Your priority is your children. Your job is being their mother.
If your boyfriend isn't treating your daughter appropriately and kindly, then you have two choices. Take parenting classes with him. Or, end the relationship. Completely.
We do not ignore our children when they act out, throw tantrums, or speak rudely. We don't turn our backs when our kids go through normal (though unacceptable and often quite exhausting) phases.
Instead, we parent. We teach. We discipline, which means we set boundaries and standards, and model appropriate behavior and responses to situations. And we punish, appropriately. Not in anger. Not with outrageous punishments, but with clear, calm, sensible consequences. If your child is throwing a tantrum, you remove her to a safe quiet place and don't respond to her tantrum. When she calms down, you speak to her quietly at eye level. If she talks to you in a rude or impolite way, you don't respond until she says something polite. You teach her how to speak to adults and other children, and if she can't control her language, she won't be allowed to have a friend over or go to a friend's house, or to the playground, until she learns to speak politely. And if you live with a partner, he must agree with your parenting rules. He can't make his own decision to ignore one kid due to a tantrum or fresh talk.
If your boyfriend is not willing to treat your child kindly and fairly when she pitches a four-year-old tantrum, what happens down the road when she acts out like most 13 year old girls will? You need a mature person in your relationship who's willing to either step out or get out. You're not a 16 year old girl. You're a MOTHER. Be one.