Learn English – What’s the meaning of “bit-h” here

While some of the definitions and idioms mentioned/linked are relatively close approximations, they all miss the specific idiom "in too deep", referenced here: (to) be in too deep

This generally references being involved in something so thoroughly that you become unable to extricate yourself (you can't escape); it is meant to allude to the experience of being in a mud bog or shallow quicksand (not enough to bury you, but too deep to escape), or to wander into a cave or woods so far that you no longer know how to return.

Note this is not precisely the same meaning as being "in deep water" or "being thrown into the deep end"; these latter two idioms indicate overwhelming challenge or great difficulty, or being forced to face a challenge beyond your experience or comfort.

As mentioned by Maulik in his comments the word out in such contexts means somewhere away from a position.

However for this particular example it has a very specific meaning. In fact Out in "Out West" does not has its individual meaning.

Out west here is used as an Idiom that refers to the "Western part of the United States". You can find the reference here.

It usually refers to the states of California, Idaho, Nevada, Arizona etc.