Turkic peoples like the Kazakhs, Uyghurs, etc. and Mongols are ethnolinguistically different, but they share a ton of cultural heritage and history. Both have a mix of east and west Eurasian features.
The Oghuz Turks first migrated westward into Central Asia, bringing their entire herds with them. A Persianized branch known as the Seljuk Turks then continued further west into Persia, defeated the Abbasid Caliphate and conquered Baghdad, then defeated the Byzantines at Manzikert to move into Anatolia. Genghis Khan then began his campaigns of conquest, and his grandson Hulagu Khan later sacked Baghdad. The Mongol empire ruled over primarily Turkic subjects, and subsequent rulers and dynasties identified themselves as Turco-Mongols (including the Timurid Empire, the Kazakh Khanate, and the Mughal Empire). Meanwhile, the non-Mongol Turks who were in Anatolia rose as the Ottomans, cut off from their Central Asian brethren by the Safavid Empire which re-took Persia.