You specifically are in New Orleans, I'm guessing?
We twist and turn, going round and round.
No need to worry, we can take the strain.
One meaning of crane is a machine used to raise and lower heavy weights, at the end of a pivoting arm that rotates about a fixed vertical axis. Thus it twists and turns, and can take the strain of a heavy weight.
While some of us are delicate and timely,
Another meaning of crane is a wading bird with a long, straight bill and long legs and neck. They are, by comparison to their mechanical namesakes, quite delicate indeed. (I don't know what "timely" references.)
Others can lift more than the strongest man.
I myself prefer to peacefully take notes in a diary.
You in particular are, perhaps, yet another crane: Emma Rose Crane, of The Crane Diaries series of books. I know nothing of the books, but found that at least in book one, Homecoming, she has come home to the city of New Orleans.