Weird that no one brought up anything around this:
We are former LDS/Mormon (so yeah, on top of societal and cultural programming, and having lived all their lives in a mono-mindset, there's vestiges of religious expectations likely floating around too -- though these are never expressed by us or them).
Unfortunately, getting into history of alternative family structures being sanctioned by religions is a little deep for the younger ones and the older one is seemingly OK with the situation....
I don't have kids and don't have parents who are poly, at least not openly. However, I did grow up with parents who have had a mutual male friend around our family for years. I almost kind of wish my mother and him did have more than a platonic relationship because he's a really great guy and has helped keep her sane around my crazy father for almost four decades now.
It's good to get them to talk about their feelings and to reassure them that their family isn't disintegrating and that there's a safe harbor for them. And, yeah, maybe it was even a bit selfish to think that the dynamic you're comfortable with would be instantly OK with them.
Beyond that, maybe there's a chance here for the kids to build their own relationships with your girlfriend (if she is that important and potentially lasting to your own life, that is). The lemonade that could be made from this situation is another strong female role model for your younger daughters - perhaps an auntie they can do things with that are separate and distinct from how they spend time with your wife and their mother. The more they see and experience her as an addition rather than a replacement, the better chance you have for this working, I think.
The other thing I'd suggest is seeking out the counseling, meeting groups, social networks of an area LGBTQXYZ resource. At the ages of your three youngest, it is a precious few kids who are OK with being different from their peers much less being strong enough to endure it and grow. Maybe the people they need to talk with right now are other kids their age or slightly older who can tell them some "perks" to the situation or give them some coping strategies for dealing with having to go underground a little when talking about their family experience with others.
Good luck and be strong. You've opened this box and, while things are still flying out of it, hope can be really hard to spot waiting in the bottom.