Normality is not the issue here. Sure, people get jealous sometimes, but there is no need to be in the situation you're describing. You have to be careful, because this can turn into an overprotective relationship where you may begin to want to control everything she does just so you can have her to yourself. She needs her own space, but she also needs to make you feel valued and loved. If you feel that she is neglecting you, tell her..if that fails, break it off. if you know that she's being fair, then you need to stop this. make her know that you're there for her and that you love talking to her..make sure she feels the same way. perhaps you have a history of insecurities which is making you feel jealous like this.
"We all take different paths in life, but no matter where we go, we take a little bit of each other everywhere."