The rise of the Rum-Seljuks
Originally just one breakaway part of the Turkish Seljuk tribe (there were also the Great Seljuks, the Kerman Seljuks and the Iraqi Seljuks) in Anatolia, on the former territory of Byzantium, the Rum-Seljuks later formed one of the greatest empires in the Muslim world. In OTL they were destroyed by the Mongols in 1243 (which allowed the Ottomans to get their own country, and later become an empire themselves), which didn't happen here. Sultan Kay Khusrau II continued to reign until 1259.
1225, they had conquered Crimea as OTL, which was the start of their interfering in OTL Ukraine. They also made a peace treaty with the empire of Nikaia in 1245; but since they weren't threatened by Mongols, ITTL they could demand tribute from Nikaia. Soon after this, they'd clash the first time with their great opponent, the Shahdom of Choresm, which had put Syria under its protection in 1262.
1265, the Rum-Seljuks attacked Nikaia, threatened its capital. This time they were still content to get some areas (i.e. not the whole empire). But in 1284, they'd attack the resurrected East Roman Empire again, and conquered Brussa, Nicomedia and Nikaia. (IOTL the Ottomans took eleven years for that, but they were one of many little princedoms in Anatolia then. The Rum-Seljuks, OTOH, already owned most of Anatolia...) 1292, the west of Lesser Armenia was conquered by the Rum-Seljuks.
Until the separation
While Constantinople may still have held out, the Rum-Seljuks were lucky when in 1302 the ambitious Charles of Valois wanted to become new Latin Emperor. Thrace was overrun by Catholic mercenaries, and in their panic the Byzantines called the Seljuks for help. The Seljuks indeed threw the crusaders out again, but they didn't think about giving Thrace to Byzantium back either... 1304, the Kingdom of Lesser Armenia became a tributary of the Rum-Seljuks, even without war.
Once having entered the Balcans, the Seljuks went on to dominate it. In 1310, they defeated Bulgaria the first time, taking the South with Philippopolis (Plovdiv) and the Aegean coast. Czar Michael Asen III fell in battle. 1314, Thessaloniki was conquered by Seljuks; 1319, the disputed area of Macedonia; 1327 even Constantinople fell after a long siege, effectively defeated by the hunger. The East Roman Empire had ended.
1329, the Seljuks won another victory against Bulgaria. Sofia fell to their hands. 1333, they conquered Thessalia. The remaining states in Greece started to panic, appealed for a new crusade, but with no success: Italy was divided, the HRE kings struggled with the pope or were too weak, France was at the beginning too poor and later under various kings, Hungary busy with Serbs and Kumans. Only in 1338, after the necessary organizations, a small crusade was organized mostly by small nations like Savoy, and Thessalia was reconquered. Bulgaria got Sofia back.
Separation and unification
In 1348, Rum-Seljuk sultan Kay Khusrau III died and divided his empire between his sons Kay Khusrau IV of Rum (who ruled the European possessions and about a quarter of Asia Minor) and Kilij Arslan IV in Konya (former Ikonion).