Well you do't need to doubt, it is not normal development....could you maybe show him a list of what is normal development. pbs.org has a good one.
DH didn't reaize for a while just how advanced DD was as he was never really around other kids her age, I read him what is normal for her age, and ten we went through the lists together to find out what age it was normal for a child to be doing what DD was...that was very eye opening to both of us.
Also he went to one of her ped appointments wit me and heard the ped say at 2 she was the ost advanced child she had ever sen in all her years of practice and said she was on a 4 year old level cognitively ....I also told him what the ped said the next time we went, which was that she was on a 5 year old level and a 6 year old level in some things.
Dh does not doubt at all now, which I feel very blessed for, and we both know we will face many challanges raising DD.
It has also helped that DH has started to see more children closer to her age so he realizes how different she is.
Do you think there is a reason your DH does not want her to be so advanced? Do you think he has concerns, or something in his past that makes him weary of labeling dd as gifted?
Good luck.
