There are no rules saying that someone who is 13 is too small for tampons. I know you think you are putting it in right but if it won't go in then you are probably trying to put it in at the wrong angle. Don't try to put it straight up, tilt it towards your back slightly and ask your mom to buy a smaller sized tampon, that should help
Hi there. I started using tampons with my very first period (way back when) when I was your age. They are made for all girls who menstruate, and for all bodies. They make very thin ones and this is what you should start with. I find the plastic applicator type to be easiest to use. Inserting a tampon just takes a bit of getting used to but is not hard and your body WILL accommodate that. You have to relax and just remember that women all over the world do this every single day. Deep breaths, gently push it until the end of the thicker part of the applicator and pull out the plastic part. Maybe waste a tampon seeing how it works without inserting it. But definitely the plastic applicator is easiest. Good luck and ask your mom for help too. good luck
PS: above "you can't" is wrong. You CAN!!