Thank you for writing to AlterHeros. Thank you for wording out your feelings, not a lot of people are able to do that. I think I understand your question in two parts 1) should you get angry that your girl is flirting with other girls? 2) should you feel threatened by it?
Let me tell you up front that I do not have a clear answer for you, for the simple reason that feelings are very complex, and when it comes to relationships, I personally believe that it is hard to control feelings. I think they come naturally. What is important is that you acknowledge them, like you did: for example, acknowledge that certain situations make you angry, while other situations may make you sad, or both. There is no answer to : What should you feel? But maybe you should ask yourself Why you feel the way you do; angry, sad, threatened.
If the answer is that you feel this way because you really care for this girl, then I suggest maybe you should communicate your feelings to her. I understand your reflex: if she flirts with other people, why not do it too, right? I understand that it might be confusing, or at least different, for a guy to date a bisexual girl. I have put guys in this situation before, as I am bi myself. But correct me if I am wrong, isn't it the goal of dating to be happy, and potentially share that happiness? If it truly makes you happy to flirt with other girls, or if it makes her truly happy to flirt with other girls, there is a compromise to make, but it all starts with communication.
Now, easier said than done. Here is a first step I can suggest: ask yourself a couple of questions as to understand why you experience those feelings. Are you angry because you would like your girl not to flirt with anyone other than you, guy or girl? Or are you angry because she flirts with girls, specifically? If it’s specifically about girls, try to understand why. I do not want to generalize your situation, but let me tell you about a guy friend of mine who has been in a long term relationship with a girl who is bisexual: he told me that when his girlfriend looks at other girls, he has this complex feeling which is a mix of 1) being scared he cannot provide what girls could offer his girlfriend, just because of the simple fact that he is male, 2) simple sadness that his girlfriend is looking somewhere else.
I am sorry I do not have a simple answer to should you get angry, or feel threatened. All I know is that you are allowed to feel any feelings you may be having. Maybe it is hard to control feeling those feelings, but it is in your power to control how you react to those feelings. It usually helps to put things into perspective. I know for myself that I get a bit jealous when I am dating someone who flirts with other people, but at the end of the day, they are dating ME, right? They chose me. So that makes me happy. And if I still feel jealousy, or anger, I just talk about it. Every person has their own comfort level. I have had partners that were comfortable with me flirting with both sexes, whereas other partners wanted me to have only eyes for them. At the end of the day, if you care about someone, and if they care about you, they will respect your comfort zones. Odds are, you like her exactly because of who she is, bisexual or not. So why not tell her, exactly like that: I like you exactly because of who you are. But I would like to talk about some things that make me uncomfortable, like when you flirt with other girls what does it actually mean to you? Haven't you heard, communication is sexy :).
Best of luck, stay true to who you are and what you feel,