For a finitely generated group there are finitely many morphisms onto any given finite group. It follows that if a finitely generated group has a bound on the index of its normal subgroups, then it has finitely many subgroups of finite index, so their intersection is again finite. If this group is infinite, then the intersection of all normal subgroups of finite index is not trivial, so the group si not residually finite.
Here is an example as above: it is $ (\bigoplus\limits_{i \in \mathbb{Z}} A_5) \rtimes \mathbb{Z}$, which is amenable, finitely generated, non-residually finite and contains subgroups of arbitrarily large finite index. I can provide the details if someone is interested.
Now the question I am wondering is: is there an infinitely generated residually finite group with a bound on the index of its finite-index subgroups?