A 15% bump at the levels you're likely taking about is quite normal and assumed to be the premium for getting you to take the risk. As long as the 20% is not outside the realms of reasonableness, you'll be fine. They may counter with a lower number, but if they do that they'll probably finesse that with an explanation of some sort (they're a fantastic company but cash poor, there is lots of upside potential, etc.). And hey, there's a chance that you'll get the 20% ask.
Did you do any salary research? Before my last job, I did that and felt pretty prepared when they asked for my salary requirement. There are sites out there where you can list job title, size of company, public/private/government, geographical location, etc. and get a pretty good idea.
If they said they'll get an offer letter out tomorrow, I expect that unless they're total flakes that you'll get an offer. Then go from there.