“Can general anesthesia cause brain damage?”

There is risk to any invasive procedure, but you need to trust your team.
Board certified anesthesiologist are medical doctors and have completed have postgraduate residency training that can take anywhere from 4 to 9 years depending on if they specialize. I hate to admit it but I couldn't do my job well if we did not have a bond. If you trust your surgeon, then trust your anesthesiologist, be as frank with them with your concerns as I hope you are with your surgeon.

General anesthesia does not cause brain damage. In fact, many medications used to achieve general anesthesia actually protect the brain. However, physiologic changes that occur under general anesthesia can compromise cerebral hemodynamics, including decreases or increases in blood pressure.

Short term (hours) confusion after general anesthesia is common. This is however no different than the effect of any other drugs/medications that depress brain function, e.g. alcohol. Moreover, a recent trial comparing general vs spinal (a form of regional) anesthesia in hip surgery patients ( https://doi.org/10.1056/NEJMoa2113514 ) did not find any difference in post-operative complications, including frequency of delirium (confusion), about 20% in both patient groups.

Long term damage is theorized to be possible. However, the current (limited) evidence from observational studies so far does not demonstrate a significant effect of general anesthesia on brain degenerative disorder such as development of Alzheimer disease (e.g. https://pubmed.ncbi.nlm.nih.gov/35981552/ ).

Studies in children (e.g. https://pubmed.ncbi.nlm.nih.gov/35788121/ ) also do not seem to definitively demonstrate a significant impact.

Also, please note that for many types of surgeries (e.g. head and neck, chest, abdomen ) local or regional anesthesia is very difficult or impossible to do.

Yes, side effects.