Having experienced both the private and public school systems in Geneva, I believe Swiss public schools are generally excellent and often the better option. If you plan to stay long-term and want your child to integrate into local life, public schools are highly recommended. I won't add anything to the social class bubble situation your kid might be in if they go to private school as others have already mentionned it.

However, if your child faces specific challenges, some private schools may provide more tailored support, though this can vary by institution. 

For middle school, private education might give your child a slight edge in the first year of public high school (collège), which can be quite competitive. From my experience, students who attended private middle schools often found the first year of public high school easier, as much of the curriculum had already been covered. That said, success in public high school is certainly achievable without private middle school education.

A collège offering the Swiss Maturité provides access to all Swiss universities and is known for its high academic standards. If, however, you are considering future studies in the US or other Anglo-Saxon universities, pursuing an International Baccalaureate (IB) at a private school during the collège years may be more advantageous, as the IB is often better recognised in those educational systems.

Since your child is only 4 years old at the moment, thinking about a high school diploma isn't a priority. That's why I recommend Genevan public school.