Regarding *upper* bounds, there are none. In more detail:

By genus theory we know that for discriminants with one class per genus, the class group satisfies
$$
\mathcal C(-d)\cong \left(\mathbb Z/2\right)^{g-1},
$$
where $g$ is the number of prime divisors of $d$. Obviously $d$ is bigger than the absolute value of the smallest fundamental discriminant with $g$ prime divisors,
$$
d_g\overset{\text{def.}}=3\cdot4\cdot5\cdot7\dots \cdot p_g.
$$
From lower bounds on the size of $p_g$, the $g$th prime and on $\theta(x)=\sum_{p<x}\log(p)$, one can show that
$$
d_g>g^g.
$$
Since $
2^{g-1} \ll \sqrt{g^g},
$
good lower bounds for the class number rule out the possibility for one class per genus for large $g$. Chowla used this idea to show that the number of classes per genus tends to infinity with $d$.

In 1973, Peter Weinberger showed that on GRH, no fundamental discriminant $-d<-5460$ has one class per genus, and unconditionally there is at most one more such $d$. Weinberger used Tatuzawa's version of Siegel's Theorem to deduce there is at most one such $d$ bigger than $d_{11}=401120980260\approx 4\times 10^{11}$, and sieving to eliminate the $d<d_{11}$.

In contrast, Oesterle explicitly observed that the lower bound due to Goldfeld-Gross-Zagier is not strong enough to finish the classification of discriminants with one class per genus:
$
\log(g^g) $ is $\ll 2^{g-1}
$.
Iwaniec and Kowalski observed that even the full strength of the Birch Swinnerton-Dyer conjecture, 'the best effective lower bounds which current technology allows us to hope for' would not suffice, as $ \log(g^g)^r$ is $\ll 2^{g-1}$ for any $r$. In fact, the outlook is still more bleak: Watkins observes that if the discriminant $-d$ is divisible by all the primes up to $(\log\log d)^3$ (as $d_g$ certainly is), the product over primes dividing $d$ in the Goldfeld-Gross-Zagier lower bound is so small the resulting bound is worse than the trivial bound.

Regarding lower bounds, sieving can be used, as Weinberger did and as has been done with all class number problems going back to the work of D.H. Lehmer. In resolving class number problems, beginning with the work of Stark on class number $1$ all the way up to the work of Watkins resolving all class numbers $\le 100$, there is always an intermediate range beyond where sieving is feasible and below where the deep theorems (e.g. Gross-Zagier-Goldfeld) take effect. In this intermediate range one uses instead the Deuring-Heilbronn phenomenon, in which a counter-example to GRH (real a zero of a quadratic Dirichlet $L$-function close to $s=1$) will influence the location of zeros (on the critical line) of other $L$-functions. Practical application of the Deuring-Heilbronn phenomenon was initiated by Stark in his PhD thesis, who used it to show a $10$th discriminant with class number $1$ would have to have $d>\exp(2.2\cdot 10^7)$.

These ideas can be adapted to the study of discriminations with one class per genus (unpublished). One can show that there is no such discriminant in the range $10^{67}\le d\le 10^{28000}$. Extending this interval upwards requires finding new quadratic Dirichlet $L$ functions with very low-lying zeros. What one really desires is to extend the interval downward to meet up with the range eliminated by sieving. This is technically very difficult (see Watkins' thesis for the analogous problem for class number $\le 100$) and is the main reason the work is unpublished.