How to understand "negotiate" in the sentence?
By successfully negotiating the challenges of life in a new country a student can mature and grow.

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navigate would be a better word. Negotiate and navigate are interchangeable in this sentence. 
an alternative meaning of negotiating​ = to find a way through or over difficult terrain.

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Here, negotiate means "to find a way through"

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I think "hurdle" or "surmount" are good words for "successfully negotiating the challenges of life"

* by hurdling (surmounting) many obstacles in a new country a student can mature and grow.

* to surmount difficulties