The sentence given with its translation is:

> I'm out of the woods.
>
> 山 は 越えた よ。

When I looked up the word "越えた", it was only matched with "越える" instead. I
think the meaning is similar. When I google translated 越えた, it gave me
"exceeded".

I'm unsure if this is an error in my material but it's certainly confusing me
as I'm still new to learning.
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grammar meaning kanji idioms
What is the meaning of 山は越えたよ?
So this is an interesting translation, because what is actually being said,
and the translation do not have the same literal meaning, but they carry the
same general meaning as a figure of speech.

In short, a Japanese idiom is being translated to an English idiom.

> 山は越えたよ。

Literally translates to:

> I crossed over the mountain.

**山** = mountain

**は** = topic marker. As a standalone sentence, I would use を instead, but you
can have this be grammatically correct if this comes in the middle of a
conversation. There are cases where は can replace を in a sentence and this is
one of them.

**越えた** = past tense of
[越える](https://jisho.org/search/%E8%B6%8A%E3%81%88%E3%82%8B), which as you have
found can mean 'to exceed,' but as @chocolate has pointed out, this definition
uses a different kanji: 超える. It is read the same way. 越える, as used in this
example, means 'to cross over/through.'  
*As a side note, jisho.org is one of the best online dictionaries I have found, and I highly recommend it.

**よ** = [sentence ending particle that adds
emphasis](http://www.punipunijapan.com/japanese-particles-yo-ne/) (almost like
an exclamation point, but not quite).

In terms of general meaning, the phrase means that you have gotten past some
trying challenge in your life. The meaning is very similar to the idiom that
it was translated to (I'm out of the woods), which is why I think that they
used the English idiom over the literal Japanese translation.
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山を越える has an idiomatic meaning, which means "to pass the peak situation of
something". For example, 彼女の病気の山は越えたよ(The worst situation of her illness was
over), 明日でこの仕事は山は(orを)越えるだろう(The most important part of this job will be done
tomorrow) and so on.
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