In fairness, it may be that he and your dd are like me and ds1 - as dh used to say, we are very alike, and that meant we could argue for ages, and neither of us would back down - and this is when he was around 15 or 16, so not much older than your dd.
We did eventually get past that stage - he grew up and matured a bit, and I learned to step back from disagreements with him, but it was a stage we had to get through in our own time - and I know it pissed dh off mightily! and he hated having to be the one who acted as the referee.
He used to try to tell me how it was winding him up, and what he thought I was doing wrong, but to be honest, it didn't help much - ds1 and I had to get through it ourselves. We did get through it, and the same stage was much easier with ds2 and ds3.
I know how frustrating it is, but maybe you need to step back, have a few deep breaths and let them grow through this stage themselves. Are they very similar characters, because that can make things hard?