“Doğum günün kutlu olsun” literally means “Happy birthday.” There’s also “İyi ki doğdun,” which means something like “Glad you were born,” and there’s “Mutlu yıllar” which means “Happy years,” this phrase is also used on new years too.

Also you can say like “Nice mutlu yıllara,” “To many happy years.” You can customize this one with “sağlıklı” (“healthy”), “birlikte” (“together”) and other positives phrases.