4 year old, crying tantrums.

My fiance has a 4 year old daughter who we have full custody of. She never see's her real mother and she actually calls me mom, and thinks of me as her mother.
Her and her older sister (6) have lived with their grandparents for the last 1 1/2 years and recently moved with us 3 months ago.
The problem is, is that she cries, actually screams, about anything she doesn't get her way.
Whether it is a toy her sister is playing with and she decides she wants, to going to bed, or even if we change the radio station in the car and she doens't want us to. She will even cry if she doens't get her hair washed first in the bathtub. She will cry and eventually starts screaming.
At times I wonder if she may have "bi-polar" disorder, but feel as though in a 4 year old, thats not possible.
She will be screaming because she is didn't get her way about something, and right in the middle of screaming, look up at you and say something like, "i really like the shirt you're wearing", or "i really had fun at the park", and then at that point be completely fine until the next tantrum.
She can be the most loving child one minute, then 30 seconds later be screaming because you've asked her to do something she doesn't want to or because she didn't get what she wants.
The problem is when we set a consiquence if she doesn't do what we ask. For instance "we aren't going to go out for pizza, if you don't stop crying", well it isn't fair for the 6 year old, so therefore we go, and she got to throw a temper tantrum, and also still got to go out for pizza.
My fiance (the actual father of these children) allows the behavior. If I say ok, you take the 6 year old, and I will stay home with the 4 year old (because she didn't stop crying), he will say something like "well she stopped now, so she can go".  
I don't know what we need to change or do to stop this behavior in her, but I need some advice. PLEASE!!!