but aren't quantum computers much more powerful than just
double-powered classical computers?

Yes. A universal quantum computer with only 100 qubits (12.5 quantum bytes) can find the ground state of a matrix with $2^{200} = 10^{60}$ elements. Assuming Moore's Law could continue forever (which is not true due to physical limitations), it would take longer than the age of the universe (13.5 billion years) for the "doubling of transistors every 18 months" to bring classical computers to what a quantum computer with one quantum gigabyte can do, for certain problems.

More interesting question is, is there even a way to improve the power of
quantum computers?

There have been proposals for exploiting certain types of phenomena that would lead to devices even more powerful than quantum computers, but in all cases quantum computers would be a special case of such devices (just like classical computers are a special case of quantum computers, they are quantum computers that just only use classical gates and inputs that are not in any superposition). It is hard enough to build a quantum computer, so building the more generalized devices would be even harder.

1I think Moore's law might be the result a complex interaction of various non-technological, mainly economical effects, too. Like this: the need to have faster CPUs, the invested stock into the CPU development, the length of the timeline until a CPU plan becomes actually manufactured CPUs in the stores, and so on. – peterh - Reinstate Monica – 2018-06-17T12:29:41.353