I understand that the process was not quick and it consisted of suppression of minorities, Turks had greater economic and political opportunity. However, why did this only happen in Anatolia (particularly the Eastern part)? Why didn't this happen in Persia where the Seljuk Turks ruled for over a century or in the Balkans/Mesopotamia under the Ottoman Turks?
I'll start off my answer by pointing out that significant parts of Iran and Afghanistan were in fact Turkified; Azerbaijan, parts of Fars, northeastern Iran, and northern and central Afghanistan. The main reason this happened, I would argue, is that the climate and habitat of these regions were (similarly to Anatolia) suitable to Turkish nomads. [I'm not too knowledgeable on Ottoman rule in the Balkans, so I'll leave that to someone else.[Though I would hazard a guess that Ottoman government was considerably more sedentary than their earlier Turkic colleagues and so were less concerned with the settlement of migrating Turkic tribes.]]
The reason for this suitability lies in the mix of mountainous and flat terrain. The Turkic nomads who invaded Iran from Central Asia, and eventually moved further into Anatolia, practiced vertical transhumance. This involved the seasonal migration of the nomads' animal herds: down into the valleys/plains during the winter, and up into the lower reaches of the mountains during summer. The reason for the noticeably smaller degree of Turkification in Iran and Afghanistan compared to Anatolia is due to the presence in Iran/Afghanistan of competing groups. Kurds, Lurs, Pashtuns and many other Iranian groups already engaged in forms of nomadic pastoralism and so would have provided competition for space with the incoming Turks. Anatolia on the other hand, was populated mostly by urbanised, sedentary Greeks, Armenians, and Syriacs. There likely were some native nomadic groups, but certainly not to the same degree as Iran and Afghanistan. Consequently, the migrating Turks faced relatively little competition during their settlement of Anatolia.
There were obviously other factors involved, such as the power vacuum in Anatolia after the Battle of Manzikert as the regional nobility fled to Constantinople to take part in the Byzantine intrigues (pun definitely intended) to select a new emperor, or in the remarkable ability of these Turkic tribes to assimilate Islamic and Christian ideas into their pre-existing pagan-influenced governing ideology. However, to reiterate it is the suitability of the terrain that I find to be the most important factor in the Turkification of these regions.