"My sister became a mother."
Translation:Kız kardeşim anne oldu.
Yes it is wrong. English almost always requires you to use a/an before singular countable nouns but in Turkish there is no such an obligation therefore we may not prefer to use it if it doesn't make sense. In this sentence it doesn't make sense for us to use 'bir' because there can be only one mother for a child. This is simply one of the logics that exists within a specific language and cannot be translated to other languages so it should be learned as it is, but as far as I know many other languages such as Spanish and German have this same logic with Turkish as well.
A similar thing occurs when we talk about professions. Say you'd say I'm an engineer. in English while you'd say Ben mühendisim in Turkish and Soy ingeniero in Spanish.