In the second chapter of string theory book by Green-Schwarz-Witten, the advantages of string theory over others is discussed. It is stated that the higher-dimensional analogs, with the action proportional to the world-volume of a "brane" object, are not perturbatively renormalizable QFTs. Their action is as follows: $$ S \sim -T_p \int d^{n+1}\sigma \sqrt{h}\, h^{ij} (\sigma) \partial_i X^\mu \partial_j X^\nu G_{\mu\nu} (X) $$ From their discussion it follows that they are referring to the "superficial" renormalizability, with the power-counting method. I am unable to understand the details of this argument.
The question Renormalizability of the Polyakov Action seems to provide some answer for this. However, it looks like this argument is substantially based on the fact that metric in the target space (which is actually not a dynamic variable) is not constant, which is not the case, say, when it is a usual Minkowski space. In this case, the considered terms that ruin renormalizability do not appear in the expansion of the action. Does it mean that in case of Minkowski target space the problem is absent? Or can one see the similar problem from the terms that come from the expansion of world-sheet metric?
