From the post-Islamic era, the Uygurs and Karakhanids developed literary Turkic, laying the foundations for literary Turki.
This was further developed during the Seljuk, Timurid and Ottoman era's.
These states had created world powers, had major world cities and a large settled population. This helped the development of literary Turkish.
Karluk-Chaghtai Turki the lingua franca of Central Asia and Oghuz Turki the lingua franca of the late Seljuk of the Rum, Anatolian Beyliks, Kara and Akkoyunlu, Ottoman lands and Turks of Iran had alot of interaction.
The works of Ali Sher Navoi, Lufti and Yusuf Has Hojib etc were known and admired by the Oghuz Turks, as was Fuzulli, Yunus Emre, Baki etc in Turkistan.
This interaction resulted in an increase in mutual intellegibility between Oghuz and Chaghtai-Karluk Turki speakers.
Then when the Oz'beks, a confederation of Turkic tribes which swept to power in Turkistan gained influence, they influence the Turki of the region with the Kipchak Turkic dialect which did not have as rich a literary tradition and was used by more nomadic tribes. However, in Oral tradition they exceeded, the Kyrgyz Manas epic is one of the longest in the world.
Why is this important?
The Turki that Uygur Turks speak has a high mutual integibility with the Turkish spoken in Turkey/Azerbaycan/Iran due to this. While Uygur/Ozbek has a very high mutual integibility, Oz'bek Turki has more Kipchak influence making Uyghur seem more comprehensible.
To summarise.
The Turkish of Azerbaycan, Iran, Turkey, Balkans, Cyprus, Crimea, Caucauses and Turkmenistan especially the southern areas is mutually intellible.
In addition the Turki of the Khorazm province of Ozbekistan has a high degree of comprehension with Oghuz Turk speakers.
Ozbekistan, Eastern Turkistan/Xinjiang, Afganistan Turki is comprehensible to each other.
Kazakistan, Kirgizistan, Tatarstan speak the Kipchak branch, they have some mutual intellegibility.
If you were to learn Turkish of Turkey, you may be suprised to realise that you can also communicate with people in Azerbaycan, parts of Iran and the Caucauses and Turkmenistan and get by in parts of Ozbekistan and even China.
It is most difficult to understand Kipchak Turkic speakers.