This is a very common question, and probably stems from a time (40+ years ago) when undergoing anesthesia and surgery was much riskier than it is now. Modern anesthesia is very safe thanks to better technology and research. Allergies to anesthesia are extremely rare, and our rates of complications are very low.
There have been recent studies which have questioned how anesthesia affects a child's brain growth. While many studies are still ongoing, the current data available reassuringly show no effect on a child's development from a single anesthetic at a young age. More information on this subject is available here: https://smarttots.org/faq-for-parents/
There are some minor side effects your child may have after an anesthetic, including sore throat, dizziness, or drowsiness.
Your child's anesthesiologist will be happy to discuss any questions you have. Please ask!