As we know "because" is often shortened as "cuz" in informal speech. Now I hear "because of" many times a day, so it came to my mind whether I can shorten "because of" to "cuz of".
You can. Some native speakers do say "'cause of" in place of "because of" when speaking quickly or rushing. It is done without thinking.
However if you need to ask the question, then you are thinking about your speaking. Native speakers don't use "cuz" when trying to speak clearly. If you find yourself naturally reducing "because of" to "cuzov" naturally, that is fine. But it should not be done artificially or it will sound odd.
You should never use "cuz" in writing, unless you are writing a story and want to show how a character speaks. Never use "cuz" for writing your own words.