"Mandens hurtige karriere fortsattes, da han blev chef."
Translation:The man's quick career was continuing as he became the boss.
The English translation is nonsense. MAYBE you could say "the man's quick career continued when he became the boss" (not "as" he became the boss), but it's still not something we would say in English.
Quick is a terrible, terrible adjective here - swift is the most likely synonym to describe a career.