"The kindergarten teacher is running on the big square behind the children and shouting."
Translation:Az óvónő a nagy téren a gyerekek mögött fut és kiabál.
There are two verbs here, though. The English explicitly has her running happening in the square behind the children, and implies that her shouting occurs in the same situation, whereas your Hungarian her shouting happening in the square behind the children, and implies that her running occurs in the same situation. I wouldn't think it a big mistake.
I think the focus of the sentence is on the fact that the kindergarten teacher is doing all this BEHIND THE CHILDREN. Your sentence places focus on either the kindergarten teacher (It is the kindergarten teacher who is doing this and not someone else), or on her running and shouting, throwing the rest in to give some idea about where she is doing her running and shouting.
You could, possibly, leave "a nagy teren" at the end.