How do you prove to a girl you are mature enough if you think you are mature?

You should try acting mature in your conversations and actions.

If you act like that in front of her she will notice and know that you are mature.

By the way act like that everywhere so you can get used to acting that way always.

Mature people get taken seriously rather than little kids.

Be calm, don't talk a lot. Listen to her when she talks. Say nice things to her, how nice she looks, congratulate her on any accomplishments. If this is a physical thing, then go somewhere with her where you can show off your body, like a pool or the beach. If this is about intimacy, go slow and think about her pleasure and what feels good to her. Always be gentle it goes a long way. That is the way mature lovers do it. If this is about the size of your willie, then anything over three inches will work.