Hi OP - It really depends on what is normal in the context of your relationship. Within my relationship, my husband doesn't have close female friends from work who he then communicates with outside of work. If he suddenly started to do that - I would find it out of character and odd and I would struggle to accept it was normal within the boundaries of our relationship. I wouldn't like it at all and I'd tell him so. And I'd expect him to prioritise our relationship and out of respect for my happiness - dial it back a bit. Likewise if I was messaging a guy from work a lot outside of work and he expressed it made him uncomfortable - out of love and respect for him I'd dial it back too. My partner comes before my work friends.
I do have guy friends that I've known since childhood. They are like brothers to me and I've done things like gone to the cinema one on one and restaurants one on one with them etc. BUT I don't text them all the time as that would signal more invested in them than my husband. Like it would eat into time spent with my husband if I was constantly texting them. Likewise my husband has met them, now knows them, has met their partners etc at bbqs and get togethers. So it's not secretive. My husband has no reason to feel insecure and he knows if he did, I'd be open to listening and taking his worries on board etc.
Your partner seems dismissive of your feelings. I'd expect to be introduced to toe honest if this is "such" a good and close friend to your partner and I'd expect your partner to visibly treat you like his partner when in front of her.
I see a lot of cool wives type on mumsnet who say their best friends are guys or their partners are best friends with a female and they/their partner have done things like gone away on holidays with their opposite sex friends alone and it's all fine - "having opposite sex best friends is fine!"
And that's totally fine for them in the context of their relationships but it wouldn't be fine in mine. I would never be happy with my husband going on holiday alone with a woman.
So don't let the people on here make you feel bad - telling you you're controlling or insecure or have mental health issues. You aren't any of those things. I am like you and wouldn't like my partner doing this AT ALL.
So like I said - you have to judge this based on what is normal within your relationship. And it could well be that you discover what is normal and acceptable FOR YOU maybe very different from what is normal and acceptable for your partner. In which case, maybe end the relationship now. Otherwise this issue will continue to resurface and ruin your relationship over and over again. It's like a monogomist trying to make it work with a polygamist. It will never be a happy satisfied relationship for either.
Next time when dating a man - discuss your fundamental values right from the beginning so you can ensure you're on the same page. Best of luck!