Why do we use 1024 / 160 bit primes in DSA?
There are two ways to solve a discrete log problem over $Z^*/p$, that is, given $g$ and $h$, find $x$ with $h \equiv g^x \bmod p$:


If the point $g$ generates a subgroup of size $q$, use a general Discrete Log algorithm (such as Pollard Rho) to recover $x$ in $O( \sqrt{q})$ time.
Use the Number Field Sieve algorithm to attack the discrete log problem in $Z^*/p$.  NFS is typically seen as a way to attack the factorization problem; it can also be applied (with some additional complexity) to compute discrete logs.


Because of this second possible method, using $p \approx 2^{160}$ would be a serious weakness.