I have to agree with Cori B. Christian schools and public schools do differ in a lot of ways. I have went to both, and so has my son. You are going to get a lot more support in not dating and staying pure in a Christian school than a public school. And I am not saying that Christian schools are perfect, because they certainly are not. There is just more of a common ground in a Christian school, and that is Christianity, which supports staying pure before the Lord (both in mind and body) and protecting your heart.
Yes, boys and girls notice each other at a certain age, especially when certain hormones kick in. God created us this way. But we also need to protect our children's heart and prevent them from getting into things they cannot handle. We need to explain that it's normal to have feelings towards and notice the opposite sex. But we need to explain to them what to do with those feelings. We have a 15, almost 16, year old son that we don't allow to date. His focus right now needs to be on school, sports, a job, his friends and things of that nature. The minute you get involved with the opposite sex your focus totally turns to that person. I also believe, at that age, your child's feelings and heart is exposed to feelings and disappointments prematurely. And, little by little, their innocence gets taken away. At that age, more than likely, the person they "date" is not going to be there wife or husband. So why "play house", so to speak? Are they ready to marry that person? Of course not. So that means there will be lots of break ups and heartaches and things that kids should not go through at such a young age.
In my opinion, 13 years old is WAY too young for a girlfriend. What does he know about taking care of a girl and handling her feelings and her heart in the right way. I believe this relationship with this girl is doing more harm than good. What does she know about loving your son? Actually, it probably is awkward for him to hear those things she tells him. It taps into feelings and emotions that he is not ready to experience or handle.
Go with your heart and your gut. I can hear in your words that you are not comfortable with him dating and with him talking to this girl. GO WITH THAT! We only have an extremely short amount of time to impact our children's lives and to protect them. I believe that if every person on this board said that you are overreacting and you need to let him date this girl, you still would feel unsettled with that. Again, GO WITH WHAT YOUR HEART IS TELLING YOU!