I remember reading that, as Arabs in the Muslim world slowly lost power to the new Turkic Muslims, that Arab caliphs would "delegate" powers to Turkic sultans in order to keep the peace, since the former had no de facto power and the latter had no de jure right to rule over the Muslim world. If this was true, about when did this start happening?
This is true and happened between the Abbasid Caliphate and the turkic Seljuk Sultanate in 1055. The Abbasid caliph Al Qaim 'invited' the Seljuk sultan Tughril to Baghdad to act as the de jure temporal ruler after he had conquered most of Persia/Iran and northern Iraq. By that time however the Caliph had lost most of his temporal power and acted mostly as a figure head of the Sunni faith. In fact, when the Seljuks arrived in the Middle East, Baghdad (the capital) and the area around it were already administered by the Persian Buyids (who were actually Shiite), with the Caliph more of a vassal.
So to answer your question shortly, The turkic Seljuks arrived to find the Arab caliphate fractured and in decline. They then acted at first on behalf of the Caliph in uniting what was left of the Caliphate, however on the ground it was the Seljuk sultans who actually ruled and the Caliph was more of a religious figurehead.
Its worth mentioning however that the Abbasid caliphate unlike its predecessor the Umayyid was more accepting of non-Arab Muslims. While the Umayyad caliphate was largely led by and employed Arabs exclusively, The Abbasids employed Persians and Turkomens and other muslims and converts in the region.