sup everyone...Im 23 years old and i just got my first bike! I baught an 03 suzuki katana 600 of my buddy for a great deal...i love the way it looks-sounds and i love the fact that i can open her up and its not going to throw me...now before a week ago i have never riden anything like this cept a manual car and just a regular bike. anyway after the first day i took it out in my development and it took me about a min to get a feel for the clutch enough to ride...after i got going it felt great...i can take off pretty fast, shift fluently up and down, use both front and rear brakes together even when slowing down modoretly to excesively, i pay attention to road conditions and look for hazards such as loose gravel and squirrels lol, and im just listing this stuff so you know about where i am as a learner u could say...i am having trouble with some turns though...mostly right turns...i get kind of uneasy because i feel like if i lean on a right turn im going to fall off, literaly i feel like the bike is going to slip from under me but left turns feel oddly natural i dont know why...also when im completely stoped its hard for me to go right without crossing over into the other lane...today i got out of work early and i took my bike out on some back roads which had freshly paved roads, some crappy sections of uneven pavement, moderate turns and some sharp turns and for the most part i did good...i was turning without even thinking about it and the bike was just following the road...now the thing that freaked me out was 2 times i found myself crossing over into the other lane...now this happened on a   straight section i think  and i could not bring it back over quick enough...what am i doing wrong..any and all advice is welcomed...write a me a novel i dont care...cause i love my bike and i want to become as safe and comfortable riding it as posible...thnks btw i have about 2 hrs riding time total maybe less at what point does one start to feel comfortable and not so nervous?!  

The best thing you can do is to sign up for a Motorcycle Safety Course. For more than one reason. You will get invaluble instruction on everything associated with riding. Most insurance companies give a discount for riders who complete the course, and in some places, Texas for example, you don't have to take the driving portion of your motorcycle license test.
The MSC I took worked on cornering over and over until it became second nature. There is no magic number of riding hours when you become comfortable on a bike, everybody has a different comfort zone. My best advice, ( and yes I am old enough to be your father) is to sign up for a MSC as soon as possible. It is well worth the time and money and will help speed up the process of feeling confident when you ride.  

It sounds like you are trying to guide the bike by shifting your body weight. It isn't a horse. You can't guide it with your knees or by shifting your weight. You have to steer it. You countersteer it into a lean and then you countersteer it back upright. At slower speeds, you countersteer it into a lean, then steer it around the corner before countersteering it back upright.

As the previous responder advised, sign up for a motorcycle riders course. You'll learn a lot about what you are doing wrong and how to do it right. The best courses are the ones that require you to ride your own bike through the course.

regards,
Joe  

One other factor that you are feeling. Except on interstate, the roads are crowned for drainage. This means to go right, you have to lean further to the right (I suspect you are steering by lean, not by countersteer).

If you want proof, find a piece of interstate (pick it quiet), and then swing from the far left to the far right, and back. It should feel almost the same either way.  

the best thing to do is to take the course.  but, did you try pushing the handlebar down the side of the direction of your turn? it works! BUT DO IT GENTLY!!!! LOL