When I turn into a corner or tilt the bike on a slower speed it pulls the steering heavily inside the turn or tilt, so that I have to countersteer it with quite a force to hold the intended line. On a straight line at higher speeds it goes perfectly straight.
Other bikes I rode were more balanced on the slower speed turn/tilt. What might be the problem with my bike?
Bike: 2006 Suzuki GSR600
Some bikes "fall into turns" more than others. While some of this behavior might be expected, you seem to be experiencing this in a significant fashion.
If the issue occurs equally in both left and right turns, then it is likely caused by an underinflated front tire. If the issue occurs on one side only, or more severely on one side than the other, then it is likely caused by front tire wear. Of course, both trigger conditions could exist at the same time.
Thus, check tire pressures front and rear, and set according to the bike manufacturer's specifications. If the front tire is worn, replace it.
Based on the fact that there hasn't been any mods, I'd say that what you're running into has more to do with the rake and trail as designed by the manufacturer, rather than any malfunction or failure. That said, there are mods you can do to change the bike's characteristics if you so choose, although I'd recommend a new question for that.
The bike's desire to stand upright is wholly dependent on the rotating mass of the tires and wheels. Lighter wheels (common in racing) allow the bike to be more nimble, and make it easier to "flick" around. Heavier wheels make it harder to flick, but make it much more stable in a straight line.
Just like the weight, the speed at which the wheels are rotating affects the bikes natural desire to stand up. So you'll notice that the bike will be more willing to dive downward when you're going slow, than when you're moving quick. This also directly affects the amount of force you have to exert to steer/countersteer since you're essentially fighting gravity.
Basically, you're wheels act just like a gyroscope (don't know if you ever had one of these to experiment with, but there are plenty of videos on the web if you don't know what I'm talking about). The faster they're spinning, the more stability they have to fight gravity and help you move them around. The slower, the more they will be affected by gravity.
I raised the rider hight on the rear of a Ducati monster m900 and if I go too high the bike starts turning inwards when going around a corner causing me to counter steer. This is caused by issues in geometry when front fork angle becomes too steep. Lowering the rear or rasing the head solves this issue. There is a perfect balance point where everything just settles into a pleasurable riding experience. I would loosen the tripple clamp and slide the forks outwards by small increments to see if it improves.
Another thing to consider, as I don't know much specifically about the GSXR, but generically speaking, is the steering head bearings. When they go bad, you may find unsteadiness in the steering.
I found when it happened to me that essentially it felt like there was a dip in the central part of the steering, and as you steered away from center, it had to sort of "rise out" of that dip. Once over it, it was not so bad. But while on the crest of it, steering was definitely wonky.
Adenda:
I should mention, and honestly it's been a while so I can be misremembering, but if you have a center stand, you might be able to test the theory by putting the bike on the center stand, pushing down the back (perhaps have a friend do that) and lifting the front, and then just carefully turn the bars, see if the steering is smooth through the entire range of motion or not.
I would not discount the simple check on your steering head bearings as the symptoms you describe could well be the result of worn or "sticking" bearings. It is more common with ball rather than taper. Check the play in the front forks - pull and push towards you with the bike on centre stand - there should be no lay or "knocking". If there is, then tighten them up but small amounts at a time. "notchy" bearings can cause what you describe although it could be tyre pressues, frame alignment, wheel alignment BUT - if it doesn't feel right then it is probably not right !
Maybe the tires have been driven flat because the bike was always ridden straight, then maybe there is a point where you fall over that corner, and it will be hard to get back. Usually only happens with big cruisers like harleys but who knows
