I don't know Turkish either...my Arabic is pretty bad but the Arabic text on that site says Mehmed bought it from the Christians because his Muslim soldiers needed a place to pray, and that the deed of sale still exists and can be visited in Ankara - although some people claim the deed is forged. (It says this happened in 481 so I must be misreading something.) (Edited to add: it was a mosque for 481 years from 1453 to 1931) It also says Erdogan will never turn it into a mosque since Istanbul already has so many mosques...

Anyway, I can't speak to the authenticity of the Ottoman document on that page...I would say it doesn't really look like it's from 1453, but hopefully an Ottoman specialist will also stop by to comment.

In any case, all the Christian or Muslim sources from the siege agree that Mehmed simply turned it into a mosque. Once the Turks had breached the walls, Mehmed allowed his soldiers to pillage the city for three days (as was customary for a siege). They believed that Hagia Sophia must be full of treasures, so that's where they headed first.

There were thousands of people hiding there, and some of them were killed in the fighting, but most were taken prisoner or enslaved. Mehmed arrived in person in the afternoon of May 29th:

“Like the Byzantine emperors before him, he rode proudly down the Mese towards the Augousteion and Hagia Sophia, passing the piles of corpses and the ransacked houses and churches. Even Mehmed was rather taken aback when he saw for himself the magnitude of the devastation that his troops had wrought, and he is said to have exclaimed sadly, ‘What a city we have given over to plundering and destruction!’, and to have shed a tear. He did not neglect to play the role of the pious warrior of Islam, however. When he reached Hagia Sophia, he dismounted from his horse, lay face down on the earth and sprinkled a handful of dust on his turban, a gesture of humility before God, the real victor. He then entered the cathedral and instructed a muezzin to climb into the pulpit and to proclaim that there was no God but God and that Mohammed was His prophet.” (Harris, pg. 212)

On June 1, after the three days of pillaging were over, Hagia Sophia was used for Friday prayers for the first time. The seat of the Greek patriarch moved to the Church of the Holy Apostles, but that church was already in ruins and there wasn't much chance of fixing it now. The site was abandoned and eventually Mehmed built a mosque there too (Fatih Mosque).

I suppose it's possible that the Ottomans purchased land and buildings from the patriarch at some point, but that certainly didn't happen with Hagia Sophia in 1453.

Sources:

Jonathan Harris, The End of Byzantium (Yale University Press, 2012)

Marios Philippides and Walter K. Hanak, The Siege and the Fall of Constantinople in 1453: Historiography, Topography, and Military Studies (Ashgate, 2011)