Playing as the Byzantium Empire, and while I've gotten back Asia Minor, my infamy is through the roof and seems to be moving down at a snails pace. Any way that I can get it moving down faster so that I'm not killing off the next half century bringing it down?
Why is your infamy high with Byz taking Asia Minor? You have cores in all provinces and you simply need to declare reconquest wars to get them without any infamy penalty. If you really took only Asia Minor and Greek/Bulgarian lands, then you shouldn't have any infamy.
My guess is that you declared one war and fought everyone that got involved. So, if you want to fight Turkish minors, check which countries will get involved (alliances and guaranties). Then declare separate Reconquest wars on each of Turkish minor, or simply take some and then wait cease-fire to pass and declare new war.
But if you are already stuck and don't want to play again, then buld up your culture and hire Diplomat. Reduction of Infamy is directly dependant on your ruler's Diplomatic skill and that is the base reduction level. Keep investing in Goverment tech (which you should anyway) to get abilty to build Embassy for -0.4/year. Embassy exists only in non-core province, so take one non-core and start biulding there immediately, so it will exist for ~49 years.
Level 5 or 6 diplomat + Embassy + ruler is already > 1/y, maybe even >1.5/y and that's best you can get early on. Depending on national ideas you may get events for reduction but that's not steady. I don't suggest changing goverment types early on as Empire is good for war and spreading.
Check in your main screen (the shield) to see if there are any of your recently conquered territories that you can release as vassals. That's the fastest way to reduce infamy.
Another (much slower) way is hiring a diplomat advisor, which reduces infamy by 0.05 points per star per year. Create him with your culture points so you get a 4+ star one and get him, permanently if you're a warmonger.
Two additional ways to burn infamy:
Switch to a republic and elect Diplomatic leaders. Each point of leader diplomacy reduces 0.1 infamy per year. Leading low diplomacy leaders to die in combat can speed up this process.
Complete missions directing you to form a royal marriage, vassalize another country, or repair relations with a country.
A short-term measure to reduce the impact of infamy is to switch to a Despotic Monarchy, which increases your infamy limit by 10.