The language family that includes Turkish, Uzbek, and Kazakh is the Turkic family. These languages are spoken across a huge Eurasian area, stretching from Turkey and the Balkans through Central Asia and into Siberia and western China.
Turkic is a family rather than a single language. Turkish belongs to the Oghuz branch, Uzbek to the Karluk branch, and Kazakh to the Kipchak branch. Their differences can be substantial, but they share historical ancestry and several structural features, including extensive use of suffixes, or agglutination. Many Turkic languages also show vowel harmony, although Uzbek has weakened this feature through strong historical contact with Persian and Tajik.
The family is sometimes incorrectly grouped with Mongolic, Tungusic, or Uralic languages under the old “Altaic” hypothesis. Modern linguists generally do not accept a demonstrated genetic relationship among those families. Similarities can result from geographical contact and shared cultural history. “Turkish” names the language of Turkey; “Turkic” is the broader family label.