Subgraph of G whose maximum degree and minimum degree are of the same order
If you want to find an induced subgraph of "sufficient size" i.e not of constant size, then this is not possible. Consider the star $K_{1, n}$, which has average degree $2-1/n$.  Take any subset of the vertices. 


if the subset contains the central node, then the max degree equals the size of the subset. If this subset is more than a constant size, then it violates your condition.
if the subset does not contain the central node then the max degree is zero, which again violates your condition.