I'm not sure what period time you're speaking of, but many ethnic groups lived as "Central Asian horse archers." These diverse horse people caused big problems for established, sedentary civilizations when they formed major confederacies and invaded their neighbors. While certain linguistic groups/tribes dominated these confederacies, they almost always included members of different linguistic groups who were united by shared lifestyles and greed.

Simply put, groups of nomadic steppe people who were dominated by indo-european tribes (but could have very well included turkic people and others) began spilling out of the steppe a VERY LONG time ago (between 2000BC-1500BC). After they conquered/peacefully mixed with sedentary people and saw the benefits of sedentary life many indo-europeans slowly gave up their nomadic lifestyle and settled down. For reasons we don't entirely understand, but which may have had to do with military superiority, indo-european languages became EXTREMELY widespread over the millennium.

As Indo-European people GENERALLY became sedentary over the millennia it seems that Turkic people began migrating west from the steppe around China and settled in areas formerly inhabited by Indo-Europeans.

However Turks were never and still are not the only steppe people, and while turkic people came to dominate large swathes of the planet, there was always another steppe people waiting to usurp them. Indo-european steppe people remained (and arguably still remain), the Mongols ruled over turkic tribes and sacked turkish cities and some groups (like the Huns) are linguistically mysterious.

We don't know exactly why indo-european tribes dominated at one time and the Turks dominated at another because these groups generally didn't keep the best written records when they were coming up, but historians have suggested personalities, localized climate change and other potential explanations.