Learn Hardcore Russian: Её глубокий вздох сказал мне больше, чем длинное объяснение. - Her deep sigh told me more than a long explanation.

Russian её can also mean her as an object pronoun, but in this sentence it comes directly before a noun (вздох), so it clearly means her = belonging to her.

A useful thing to remember: as a possessive word, её does not change its form:

её брат = her brother
её книга = her book
её друзья = her friends

Why is глубокий in that form?

Because adjectives in Russian agree with the noun they describe.

Here глубокий describes вздох, and вздох is:

So the adjective also has to be masculine singular nominative: глубокий.

Compare:

глубокий вздох — masculine
глубокая пауза — feminine
глубокое чувство — neuter
глубокие мысли — plural

Is глубокий вздох a normal Russian expression?

Yes. Глубокий вздох is a very natural way to say a deep sigh.

Russian uses глубокий not only for things that are literally deep, but also for breaths, sighs, thoughts, feelings, and so on. So this phrase sounds completely normal.