Hypothetically, if a 16 year old were to get drunk four or five times would they experience permanent brain damage or is that just applicable to binge drinkers and more avid users?

One of the breakdown products of ethanol is formaldehyde. This is what we use to preserve autopsy specimens. When you drink to excess you have a greater short term risk of dying in a car crash or other accident. The injury to your brain is long term and hard to detect by simple methods. If you plan to use your best brain for your lifetime you should take steps to keep it healthy.

Alcohol is a poison that never did anybody any good, ever, so stay away from it. It will destroy your brain cells, damage testicular cells that make testosterone, facilitate development of cancer in many organs, cause hepatitis, cirrhosis and pancreatitis etc. There is no safe level of alcohol intake. Quit while you are ahead. What you are doing is unhealthy and illegal.