This story originated and was first publicized by the Jewish Christian and long time Christian Missionary to the Jews, Ludwig Schneider, on page 18 in the March 2008 edition of Israel Today Magazine. Here is the link.
He states clearly that this is found in Sefer HaGematriyot of Rabbi Yehuda HaChassid (Yehudah ben Shmuel).
I have the sefer and to date have been unsuccessful in locating the citation he refers to. When I have questioned anyone, Jew or otherwise, as to the precise citation, no one seems to be able to produce an exact page or parsha. The book is long and the gematriot and other tzerufim are numerous. I will, bli neder, continue to go through the sefer and if and when I find the citation, will report back.
Schneider says this is a calculation relating to a 400 year empire that would rule over Jerusalem. The Ottoman Turkish Empire, which was formed by Osman I in 1299, didn't exist at the time of Yehudah HaChassid's death in 1217. Osman I lived from 1258-1326. So whatever reference Rabbi Yehudah HaChassid might have made is not about the Turkish Empire. It would be generic. If the citation is accurate, the association to the Turkish Empire is speculation by Schneider.
For the time being, the time line stated by Schneider is the following:
He says, Allenby conquered Jerusalem from the Ottoman Turkish empire on the first day of Hanukkah, December 9, 1917 (24 Kislev 5678). He counts back 400 years which he says corresponds to 8 Yovelot (Jubilees). There is a Torah sourced basis for this 50 years per Yovel calculation. That places the year at 1517 (15 Kislev 5278) which corresponds with the approximate time frame that Selim I of the Ottoman Empire conquered the area of the Middle East which included Jerusalem from Shah Ismail of the Safavid Persian Empire.
The significance of the removal of control of Jerusalem from the Muslims, the Ottoman Turkish empire is based upon Yechezkel 46:17.
וכי־יתן מתנה מנחלתו לאחד מעבדיו והיתה לו עד־שנת הדרור ושבת לנשיא אך נחלתו בניו להם תהיה
"But if he gives a gift from His inheritance (meaning Yerushalayim, the undivided Capital of the land of Israel) to one of His servants, it shall be his until the year of liberty; then it shall return to the prince. His inheritance shall be only his sons'; it shall belong to them."
The 'Prince' (Nasi) being referred to here is an abbreviation of 'Nitzutz Shel Yaacov Avinu". It means the entire Jewish people as a single entity, the idea of 'Tzibbur'. And so Yovel (Shnat HaDror) starts to be counted either when the 'possession' begins to be removed from the servant (the Islamic Caliphate), which would be 1917 or when it actually comes into the possession of the the nation of Israel (the children), which would be 1967.
Schneider points out that from 1917 until 1967, another Yovel of 50 years, Jerusalem was like a "no mans land" with control shifting between Britain and Jordan. During that time period, the modern state of Israel was established on May 14, 1948 (5 Iyar 5708). And then, on June 7, 1967 (28 Iyar 5727), Jerusalem was reunited under the Jewish state's control during the Six Day War with Rabbi Shlomo Goren actually ascending the Temple Mount and blowing shofar from the Dome of the Rock.
To complete this picture, Christians also try to connect this with the blood moon prophecies found in Yoel 3:3-4, 4:1-15. These particular prophecies speak to days of judgement for the nations of the world. Yoel 3:3 in particular is classically associated with the beginning of the first of three wars that comprise the Wars of Gog u'Magog. (See Sefer Avkat Rochel by Rabbeinu Makir ben Abba Mori, Book 1, Signs 1 through 5. It seems to be associated with the detonation of the two Atomic bombs at Hiroshima and Nagasaki which began on August 6, 1945 (27 Av 5705). The "Date Palm trees of smoke" corresponding with the "Mushroom Clouds". The shape is the same.
As is known, following shortly on the heels of the reestablishment of the modern state of Israel, a blood moon cycle started on April 13, 1949 (14 Nisan 5709). This also happened again just before the reunification of Jerusalem on April 24, 1967 (14 Nisan 5727).
As is common knowledge, we just completed another blood moon cycle this last Sukkot, September 28, 2015 (15 Tishrei 5776).