a turn: the place a person holds in a fixed order of people waiting to be called forward
Both times the word carries this sense it stands in the same phrase in the book of Esther: a young woman's turn came to go in to the king. The women gathered into the palace were kept in a set order and sent for one at a time, each after a full year of beauty treatments. The word names her slot in that line, the moment the rotation reached her.
The idea behind it is circular movement. The verb it comes from (תּוּר) means to range over a place, going around it to see what is there, and from coming around again you get a turn that comes around to each person in order. Later Hebrew never lost this sense: the everyday word for a queue, or for whose turn it is, is still this word today.
Because Esther is set in a Persian court, readers sometimes assume that a word this administrative must be borrowed from Persian. The scholars who catalogue the genuine Persian words in Esther do not include it. It looks like ordinary Hebrew, simply at home in the later style of this book.