The phrase "achieved supremacy" is correct and usable in written English.
It can be used in contexts discussing dominance, control, or superiority in various fields such as politics, sports, or business.
Example: "After years of strategic planning and execution, the company finally achieved supremacy in the tech industry."
Alternatives: "gained dominance" or "attained superiority."
Erdoğan did not move against the military until he believed that he had achieved supremacy over it.
Such is the logic of the country — an administration that has achieved supremacy in the creation of idle jobs.
Spain and Portugal tussled over the so-called Banda Oriental (Eastern Shore) until Spain achieved supremacy in the 1770s.
Roman law strongly influenced the law of contracts and torts; canon law achieved supremacy in the field of marriage; and combinations of Germanic, feudal, and Roman traditions developed in matters of property and succession, or inheritance.
By October, the Manchus had achieved supremacy in northern China, and a ceremony was held at the Forbidden City to proclaim the young Shunzhi Emperor as ruler of all China under the Qing dynasty.
Some will be commercial versions, hoping to drive out rivals and achieve supremacy.
The White Star Line wanted to achieve supremacy on the world's leading intercontinental link, which – then as now – connected southern England with New York.
Men are "hardwired" to cheat, ignore their wives, suspect infidelity, overspend, fail, love money, pursue women and achieve supremacy in the workplace.
The same people grumbling about Neckbeard Deathcamp's (intentionally) watered down war metal are probably big fans of bands who drape themselves in bike chains and smear black facepaint all over themselves and grunt about achieving supremacy through powerlifting, so really, who's the joke?
If you aren't rocking the face, you can still achieve supremacy using the "butter face" theory by focusing on body parts instead (everything's attractive "but his or her face").
Recorded music achieved total supremacy: the only notion of performance was in the skill of the deejay.
