The secret to doing well as the OE is avoiding war with the timurids at all costs, and keeping at least one ally you can reliably depend on to help you in major wars - either golden horde or the mamluks, and preferably both. Vassalising both the anatolian minors you start off at war with is preferable but hard, and don't be afraid to give up a province to the Timurids at the start for peace, if needed. DO NOT release vassals, if you give away a province it will almost always revolt back to you, or one of your weak neighbours. Always take military drill as your first NI and national bank as the second; as tempting as it is to go for the economy first you absolutely need drill to counteract the horrible WE you will have early on.
Once the initial war is over and you've got your strength back, then you can take out as many of the christian nations around you as possible. Only take greek / bulgarian provinces, or any other provinces you have cores on (such as Serbia via mission etc), never take wrong culture, wrong religion provinces without cores as the revolts will kill you early on. A strong navy will help vs Venice / Genoa. Focus on anatolia and greece first, then go for the killer blow against the timurids during a successon war - your aim is to take their Iraqi (and any turkish) provinces as they make an excellent staging ground for moving into Arabia. Preferably the WE will make the Timurids implode at the same time. Don't betray your allies until you're absolutely sure you can withstand the largest 2 alliances in Europe attacking you at the same time, since the catholic AI hates the OE with an insane passion.
I've had success getting the OE into the 1500s with a 100k army and controlling all of arabia, anatolia, greece, the balkans and northeast africa, but in both games I've come unstuck against a 300-500k army christian alliance lead by an insane HRE (Austria in one game and Bohemia in the next). Though I could beat them into a white peace every time (slowly...) the constant warring with no gain for decades made the game utterly unenjoyable.