Star Wars Death Star question?

So I have what I believe to be a very valid Star Wars question about both of the Death Star's. So at the end of Revenge of the Sith, you can see Darth Sidious and Darth Vader overlooking the construction, which is in its early stages of creation. Now it isn't until 20 something years later that they finally use it's super laser to destroy a planet. Now fast forward to Return of the Jedi and they're constructing the 2nd Death Star. Now bear in mind the time lapse between Revenge of the Sith and A New Hope was about 20 years, give or take a few. The time lapse between A New Hope and Return of the Jedi was 5 years, at max. Likely a little less.

Now unless they were working on the 2nd one while the first one will still in operation (which I find highly unlikely), how in the name of hell could they have even started and already made the 2nd Death Star operational in as little as 5 years, while it took 20 years or more for the first one to become operational? And even if the first had been operation before A New Hope, I find it very unusual they hadn't tested the weapon on a planet already.