It’s very difficult to need both men and women to satisfy your needs, and yet want a monogamous relationship that will leave you wanting in one way or another.
Your girlfriend senses that she won’t ever be enough for you, and is setting you free to satisfy the needs you have for men. She cares about you enough, deep down, to let you be yourself.
However, if you do have sex with other people (men or women), she is most definitely going to be hurt. I know that you don’t want to hurt her — or lose her, but your bisexuality coupled with your need for sex from both genders, is not a prescription for monogamy. You can’t have everything, so you’re going to have to choose.
If you sacrifice your sexual needs for men in order to stay faithful to your girlfriend, you’re cheating yourself. If you sacrifice your girlfriend’s security in her relationship with you, you’re cheating her.
What may work best for you is a relationship with someone else who is also bisexual. You’ll still have the issues of satisfying each other’s needs for a different gender sexual relationship, but you’ll come to the table with what I like to call “matching luggage” and you may have an easier time making compromises with another bisexual than you do with a heterosexual woman.
Let me know if that helps.