How long have you known this girl? And how long have you two been exclusive?

She's not ready to commit to you and doesn't feel safe enough or confident in your relationship to cut off other people. Frankly, she's just not that interested or doesn't feel that she knows you well enough/connects with you enough to commit to you 100%.

Her response after you read the texts was reactionary- she only said that in a moment of worry to try and not lose you either. It wasn't a sincere desire to be exclusive with you.

She just sounds extremely immature.

You should never agree to exclusivity with someone if you don't trust them and want to actually be in a relationship with them. She never should have made a promise that she wasn't ready to keep.

If she wants to date casually and keep talking to other men, that's fine but she should be 100% transparent about that and risk losing you as a potential bf.

It sounds like she's just acting out of fear and refusing to be honest about what she wants and what pace she's comfortable with. If you haven't known her for long, I don't blame her for not wanting to commit when she doesn't know if you're trustworthy and if you're someone she actually wants to be with. Her error is acting out of fear and being deceptive about what she wants and what she's ready for.

Now you need to not replicate her immaturity and naiveté and see these red flags for what they are. Her actions speak louder than her insincere words. She sounds way too immature to be a good person to date, and in my experience people who keep a "roster" of options are insecure people who need lots of validation and they don't make good partners. I'd cut things off and block her immediately.