Im not sure if this is exactly what you are looking for, but try searching google for the "dimension growth conjecture". This is essentially the conjecture that $N(X,B) \ll_{d,\varepsilon} B^{dim X + \varepsilon}$ for any projective variety $X$ of degree $d$.

I think this is now known for any variety whose degree is not three. This follows from work of Heath-Brown, Browning, Salberger, Marmon and others. Also, Salberger has recently announced a proof for the case of degree three, however the implied constant is not uniform with respect to $X$.

These results are generally proved using a higher dimensional analogue of the determinant method of Bombieri and Pila, in particular Heath-Brown has developed a $p$-adic version of the determinant method that has proved fruitful. This might also be used to give you the kind of result that you are looking for.

Sorry I cannot give you explicit references and more details, but I am just about to go on holiday!

Im not sure if this is exactly what you are looking for, but try searching google for the "dimension growth conjecture". This is essentially the conjecture that $N(X,B) \ll_{d,\varepsilon} B^{dim X + \varepsilon}$ for any projective variety $X$ of degree $d \geq 2$.

I think this is now known for any variety whose degree is not three. This follows from work of Heath-Brown, Browning, Salberger, Marmon and others. Also, Salberger has recently announced a proof for the case of degree three, however the implied constant is not uniform with respect to $X$.

These results are generally proved using a higher dimensional analogue of the determinant method of Bombieri and Pila, in particular Heath-Brown has developed a $p$-adic version of the determinant method that has proved fruitful. You might be able to use this to get the kind of result that you are looking for.

For an overview of these results I would recommend: http://www.maths.bris.ac.uk/~matdb/preprints/pila.pdf