Hi, you are not messing it up.
When people are emotional and fighting, they are in a mindset, that is set in fight.
So whoever gets in the middle is drawn into the fight.
One question is the way they fight, if that realy is "only" a differnent way to communicatie between two persons, that
love to fight, you can let them and just ask them to do the fighting in an extra room, for example, so that
you need not be part of it.
So all 3 of you find a way, where none of you is hurting. At present you hurt, because
you do want them to be happy, which for you means no fighting. To then fighting might
make them happy.
So say, if you (wife & daughter) want to yell, go to the kichen (f.ex.) and close the door, untill you
are finished and then reasure me (you), that all is ok. Because they have to understand, that
you need (!) that reasurance.
If they are realy fighting (without having fun about is), that is different.