Is there a method to speed up growth during puberty or push a child's first period for later in time without pills?

Puberty: Hormones play a role in all of your questions and most outcomes will be determined by genetics. No reason to try and change your child's normal development unless there is a medical condition.

Not really: If a kid has a poor diet or poor sleep habits, correcting them can help things along. Pills would not be any real benefit, as the hormones they contain are broken down during digestion. Rarely, an endocrinologist can plan a short term of hormone injections that could push a kid to develop faster.