That is an old myth now.
Synthetic oils are simply Engine Oils with extra properties. Many people think Synthetic oils are like “Acid” and it will destroy the engine. Its wrong concept and is not like that.
If you are already using mineral oil (which normally comes in 10w-30, 10w-40, 15w-40 and 20w-50. That means you are not feeding the recommended oil to your car.
Honda recommends 5w-30 of SL, SM grade.
So you better shift on 5w-30 and keep the change interval like 5~6K depending upon your driving habbits.
5w-30 comes in semi synthetic and in fully synthetic also.
Prefer the one which is fully synthetic.