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!