In spite of having quite good working conditions, I'm looking for opportunities to grow up in my career.
Does it sound ok? I'm not sure about the second part of the sentence, especially the phrase "to grow up in my career".
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The phrase "grow up" is usually used for children growing up into adults. Sometimes an adult might be told "Oh grow up!", meaning that the speaker thinks they're behaving like a child. You definitely don't want to use that phrase in your sentence.
You can talk about wanting to grow in your ability or skills. Usually we say "advance" in your field or career, as Stephie suggested. Post graduate degrees are sometimes called "advanced degrees", so advance has a sense of moving toward a higher level of skill or knowledge or responsibility.
It makes sense, but the wording is awkward. You could try being more specific about what you hope to do. For instance:
I am hoping for an opportunity to improve my interpersonal skills
I hope to develop my understanding of the industry
Sentences like this make it clear that you know what you want to achieve, and give a clearer idea of you as a person and how you could become more valuable over time.