One time, as the English, I launched a crusade to try and get the crusader goals before high. The Byzantines didn't like letting me through and declared war... and ran back to the keep... My army was big enough to make the AI run away.

So to make a long story short:

The Byzantines kept attacking the provence my army was in but my crusade would keep moving forward. It seems like a reoccuring theme in alot of my games that Constantinople is left undefended(probably due to it being a bottle neck.

So guess what?

I ended up accidentally capturing it. I decided to be merciful... to the Turks and stripped 10K worth of buildings out of Constantinople. And that was the end of them. The Turks eventually took over until the horde killed them all.

You could always kill Constantinople, that's pretty much an autowin against AI controlled Byzantines.