I guess it is normal, plenty of people get that way. But that's more to do with you than it is to do with her. She has her own life and she is free to talk to whoever she wants to, which I am sure you're aware of.
Why do you get jealous? Figure that one out.
Because if you trust her then it's clearly got nothing to do with her being untrusting or whatever, this is all on you. Are you afraid that she will find someone better, or what? Figure out what the deal is and then go from there.
To love is to suffer. To avoid suffering, one must not love; but then one suffers from not loving. Therefore, to love is to suffer, not to love is to suffer, to suffer is to suffer. To be happy is to love; to be happy then is to suffer, but suffering makes one unhappy; therefore to be unhappy one must love or love to suffer or suffer from too much happiness. I hope you're getting this down.