They sort of did, but there were a lot of other things going on that prevented them from driving out the Seljuks. They were fighting each other just as much as they were fighting the Seljuks!

The Seljuk Sultanate of Rum wasn’t the only Muslim power in Anatolia. There was another group of Turks there as well, the Danishmendids, who had established their own emirate around Amasya. The Seljuks had made it as far west as Nicaea, which was initially their capital until it was retaken by the Empire during the First Crusade in 1097. After losing Nicaea, the Sultanate of Rum was centred on Konya.

In southeast Anatolia, the crusaders had established their own county around Edessa in 1097. Edessa had actually been part of the Empire before the crusade, but the Byzantines didn’t seem to care about that, maybe because it was mostly in Armenian territory. The Armenians, meanwhile, took advantage of the crusade to establish their own kingdom in Cilicia.

Further to the east in Mesopotamia, there were independent Turkic states ruled by the Artuqids in Hasankeyf and Mardin, among others. The Artuqids also controlled Aleppo, further south in Syria. They defeated Antioch at a battle in 1119 that was so disastrous for the crusaders that it was called the “battle of the field of blood”. In 1122 the Artuqids captured the count of Edessa, and in 1123 they even captured king Baldwin of Jerusalem, who had come north to help Edessa.

Elsewhere in Syria, Mosul, Homs, and Damascus also had their own independent emirs and atabegs. In 1128, Mosul and Aleppo were united by Zengi, who also tried to capture Homs and Damascus. Zengi won the Battle of Barin against Jerusalem in 1137, and he conquered Edessa in 1144.

As for the Byzantines, the Komnenos dynasty had come to power through a coup in 1081, so their legitimacy was always a bit shaky. Emperor Alexios’ priority during the First Crusade was to recover Antioch. The crusaders had promised to return Antioch to him, but they didn’t, so he was preoccupied with subduing the new crusader princes. Meanwhile on the other side of the Empire, the Byzantines also had to deal with invasions from the Normans in Sicily. Not coincidentally, the Sicilian Normans were also now ruling Antioch! Alexios defeated the prince of Antioch, Bohemond I, at Dyrrhachium in the Balkans in 1106.

Alexios did fight against the Sultanate of Rum sometimes. In 1113 the Seljuks unsuccessfully tried to take back Nicaea, and Alexios defeated them again at the Battle of Philomelion in 1116. His son, John II, took Laodicea in 1119 and and Sozopolis in 1120, but John also had trouble on the western frontier: in the 1120s, the Hungarians and Serbians attacked Byzantine territory in the Balkans, and Venetian ships pillaged some of the islands in the Aegean. Otherwise, John also spent much of his reign trying to control Antioch.

In hindsight it might also have been ideal for the Byzantines, Armenians, and crusaders to ally against the Turks, but they didn’t see it that way. In 1130 the Armenians and Danishmendids allied against Antioch, and the prince of Antioch (now Bohemond II) was killed in battle. The crusaders further south in Tripoli and Jerusalem also had to deal with invasions from the various Turkic rulers in southern Syria, as well as attacks from Fatimid Egypt, so they weren't able to help out in Anatolia.

So, looking at the map it seems like it would have been easy to surround the Turks and drive them out of Anatolia. But the reality was not as neat as the map. the Byzantines and crusaders often had incompatible interests, and they didn’t always get along with their fellow Christians in Armenia who were trying to establish themselves in Cilicia. The crusaders in Antioch and Edessa faced numerous threats from the Seljuks and Artuqids in Mesopotamia and Syria, and the Byzantines also had to protect their western territories from their European neighbours.

Sources:

Ralph-Johannes Lilie, Byzantium and the Crusader States, 1096-1204 (Oxford University Press, 1994)

Jonathan Harris, Byzantium and the Crusades (Hambledon and London, 2003)

P. M. Holt, The Age of the Crusades: The Near East from the Eleventh Century to 1517 (Longman, 1986)