Hi Blossom8:
You clearly have a bright little girl there who's doing very well indeed.
First - is she at nursery? If so - why not ask them what they think is best. They'll know where she is at in comparison to the other children and the type of schools the other children have gone to. Their advice might be quite helpful.
Second - as long as it's not endless flash cards I suspect she enjoys time with you and learning. Usually if a child doesn't enjoy it - it's pretty obvious. I'm sure you're not forcibly educating her.
Third - visit the schools you are considering (both state & private sector). It's difficult on an open day - they're putting their best face forward - but if they invite you to have a wander around the shcool on your own, take them up on that. Make the time to really work out which school you like best.
My impression is this, basically most primaries do a good job. I have posted elsewhere that I have several issues with our school and would gladly move to a better one (several nearby) if we could, but I also know that basically with a bit of help/ support at home my DDs are doing well.
So if you are happy to be involved (and enjoy it) - my advice is consider the atmosphere of the private school & ask around. If the feedback is that it's a bit snobby and the staff are at each other's throats - then seriously consider the state sector, but bear in mind that you will most likely have to do some home support as well.
One key fact to look into (because it will be reported from this year) is how many pupils achieve L6 on KS2 SATs - this Level 6 figure is considered exceptional (the target in 4a and end Year 6 age 11, L5 is considered above average). If the school can support pupils with that level of achievement that's a very good sign that they can support a child that looks to be exceptionally bright at near age 3.
You can also ask about what provision they have for gifted and talented pupils when you visit.
Finally what I will say - and to take the pressure off - is no matter what decision you make there will be pro's and con's. No school is perfect. One school might offer Chinese (which you'd love your DC to learn) but your school might not. One school might have great music opportunities - with all sorts going on (choir, orchestra, lessons, etc...) and your school might not. Your choice may have had a fantastic Head Teacher, but they retire and the new guy is dreadful. Also accept that there will definitely be one or two teacher's you just don't see eye to eye with. They're too strict or not that inspirational, whatever. So there is no perfect choice and you need to know that.
HTH