They should have instead be termed as being of Afghan origin and not Turkish origin since their founder Qutb-Ud-Din Aibek was the slave of Muhammad Ghori who was the Sultan of Ghur in Afghanistan.
Mu'izz had no offspring, but he treated his Turkic slaves as his sons, who were trained both as soldiers and administrators
and from the Concise History of Islam:
"Other monarchs may have one son, or two sons; I have thousands of sons, my Turkish slaves who will be the heirs of my dominions, and who, after me, will take care to preserve my name in the Khuṭbah (Friday sermon) throughout these territories."
and back to wikipedia:
Mu'izz's prediction proved true. After his assassination, his Empire was divided amongst his slaves. Most notably: Qutbu l-Din Aibak became ruler of Delhi in 1206, establishing the Sultanate of Delhi, which marked the start of the Slave dynasty.