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It's okay to publish a book at any age. If it is a good book (or at least good enough.) Worse come to worse, if you can't publish at age 13, you can try again at age 18 or age 23. A famous writer, Pearl S. Buck published her first book at age 10, I believe.
Picturebooks are written for children ages 4 to 8 (or 3 to 8). At this time, children typically enter the emergent reader and early reader stages. Also, their attention spans are longer and they can sit still for more time. They are now ready to leave board books behind and read longer books, i.e., picturebooks.