Strictly speaking birds don't even have a cerebral cortex. That is, cortex evolved in mammals after the split with the last common ancestor of mammals and birds. So in that sense, birds would be at the very back of the ordering, along with reptiles, fish and the other vertebrates that lack a cortex. That said, birds do have a well developed pallium that seems to play a similar role as cortex does in mammals, even though it is structurally quite different. If you let pallium stand in for cortex, crows would probably do okay.

It seems likely that avian pallium performs similar computations as cortex, despite having a different implementation. The more interesting question, in my opinion, is not who has the most neurons of any particular type, but how those cells enact computations. Does the avian pallium have representations that are similar to mammalian cortex. In other words, does it solve problems in similar ways/using similar algorithms as cortex or did its early divergence allow it to find different solutions?