I am having trouble with pronouns

Grammarly is not saying your usage IS wrong, it is saying it MAY be wrong.  When you use personal pronouns -- he, she, they -- you need to make sure the reader knows to whom you refer. Someplace, one or two sentences before in the text, you need to name who he or she represents.

Because the software isn't "smart", it is asking you to look at the text and double check. In your case, you provided a name -- Louise -- and you properly used "she".

Rahul, provided an alternate that avoids the pronoun, but there is no reason (other than personal preference) to change as your usage is correct as stands.

To avoid it you can use it as a phrase:

Turning around.............

I agree with Jeff.  The program is likely prompting you to double check because the sentence isn't very clear.  This really should be broken into two sentences.  The phrase, as she turned around, is set apart by commas, meaning it is an independent clause.  That means the sentence can stand on its own without it.  If you read the sentence without it, it becomes more evident that you have too many actions (she said and gave a smirk then rushed) going on for one sentence.

“You know I don’t like to be late for my meetings,” Louise said as she turned around and gave Fanny a smirk.  She then rushed across the green lawn toward her house, which was located some fifty feet away.