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The answer depends on the factorization of $p-1$; for any factor $q$ of $p-1$, the attacker can derive $x \bmod q$ in $\sqrt{q}$ steps (by first computing $(g^x)^q = g^{xq}$, and then searching for the value $y$ with $(g^{xq})^y = 1$ using, say, Pollard Rho. Hence, if the attacker has a computation budget of $n$, and $p-1 = q \cdot r$, where $q$ is $n^2$ ...

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