OK, troubleshooting over the internet is hard. We've had threads go for weeks before we figured out what the problem was. It took 184 posts before we figured out why Ilya's clutch wouldn't work.
What helps is being precise in our terms and having a common understanding of those terms.
For example, when you say "won't turn over" what that means to me is that when you push the starter button the engine will not turn. As in:
1) there is a clicking noise, but nothing else happens (dead battery is the problem)
2) there is a grinding horrible noise (the starter gear train is AFU)
3) there is a grunting noise and you cannot get the engine to turn even by pushing the bike with it in gear (engine is seized)
Yet you say that when you replaced the fuse the starter spun (and I have to assume the engine rotated as well), so I don't think that's what you meant. This leads to confusion.
You also refer to "ohming out" the starter button and "wires 3&4". That doesn't help because we don't know which wires you checked, where they were, and if your "3&4" is based upon looking at the relay or looking at the schematic and if you are figuring "3&4" to be counted from right to left, left to right, or some clock direction.
Don't take offense; Spock once famously said "This thing you call language though, most remarkable. You depend on it for so very much. But is any one of you really its master?" and we're all at the mercy of using common terms. Unfortunately, we have to define them to have a fruitful conversation. You can do that by describing what you mean when you use a term.
Let's look at the starter and kill switch. On the right-hand side of the handlebars, there is a connector with four wires. They are Blue/White, Brown, Brown/White, and Yellow/Red. If you check the Brown to the Brown/White you should have 0 ohms with the kill switch in the "Run" position and an open circuit with it in the "Off" position. If you check the Brown/White to the Yellow you should have an open circuit. When you press the starter circuit you should have 0 ohms. Checking the Brown to the Yellow/Red you should have an open circuit until you push the starter switch; then you should have 0 ohms. This confirms that the switch block works right. The Blue/White is the brake switch; this should not be part of your problem but its function is checked between Blue/White and Brown.
You are probably checking the starter switch between some Brown wire somewhere and the Yellow/Red wire somewhere. Where is important. The Yellow/Red wire is connected to both the armature and the coil of the Starter Circuit Relay. If you have a short someplace you may be checking the resistance of that coil.
There is a "3&4" on the Starter Circuit Relay. If you check those with the relay unplugged you would be checking the armature of the relay and that should be an open circuit. However, if you are checking them in the harness, #3, the Black wire, goes directly and indirectly to all kinds of places, and the Yellow/Red wire goes not only from the starter switch to the Starter Circuit Relay but to the igniter as well.
With a passive test and the possibility of a short somewhere, you just don't know what you are checking or what the readings mean.
What this means is that it is important to isolate components when checking.
Frankly, I think you should be looking, first, for some chafing and a bare wire in the wiring loom. The switches and components are fairly robust. Start with the obvious, which in the case of the '08 is in the vicinity of the regulator/horn/overflow tank support brackets. There are some sharp edges there. There are also sharp edges by the main fuse/starter relay. There are some pictures in this thread: https://www.klrforum.com/2008-klr650...tml#post675177
and Paul Westman included some more thorough information than what is in those pictures (which are very old).
A couple of years ago I was on a multi-day ride and things went all wonky with turn signals and such. It took a while but this is what I found:
Keep coming at us with info and we will help as best we can.