Why do Western governments seem to care more about Muslims than about Christians? .
I think it is due to what our Pope called a pathology of "self-hate".
They fight for everyone else's rights except their own.
The Moslems stick to their identity, and no one can take it away from them, not even when they are in a Christian majority country. On the other hand the Christians of Europe have lost their Christian identity, without convictions and easily swayed by every passing wind, like a plant that has been uprooted.
So if they lost their identity as Christians, they do not feel any closeness to the Christians in Iraq.