1. You're a big girl now, right? Are you a baby? No? Try to act like a big girl, okay?
2. Please don't be fussy. Let's relax for a moment and calm down.
Please don't cry. Why don''t you tell me why you are upset.
I know you can be a good girl. You are a big girl now and you know how to behave.
You could say "Don't act act like such a peevish child" but those are very strong words to say to a child, I think. I recommend not using them. In my opinion, a little bit softer is better.
For example:
"Please don't act like that."
"Please behave yourself."
"Please act like a good boy/girl. I know you can do it."