You can find a lot of bikes out there for $3-$4k. You just have to decide what style of riding you will be doing. Personally I prefer the dualsport/adventure type bikes as you can hit the dirt roads or the interstate. If all you want to do is ride on the street, I'd seriously look into a SV650 if you want something a little sporty, without all the expensive plastic. I took a 2 week trip across country and it never even hinted at any problems. They are dead reliable, super easy to maintain. Bikes with plastic are a pain in the arse to change the oil if you service them yourself. If you want a dualsport type bike, I'd recommend a DR or KLR.
I don't have a picture of my current bike, which is a 2012 Triumph Tiger 800 XC, but here are some pictures of the bikes I have owned in the past.
I don't have a picture of my current bike, which is a 2012 Triumph Tiger 800 XC, but here are some pictures of the bikes I have owned in the past.