There's a slider in one of the options menus called "stability." In real life when you're riding a bike, you do a lot of things to correct the bike and keep it upright that you're not even aware of, and you can do this because you're balancing on the bike and so your mind can react to even the slightest deviation from vertical. Inputs to the handlebars of fractions of a degree, constantly act to keep the bike oriented the way you want it to be.
But in MXS, you're not balanced on a bike, so you're getting none of that input, and therefore you're not able to input those tiny corrections. So, the game has a complex system of doing it for you, but it's not set up quite right for everyone. The stability slider generally dictates how much of this automatic stabilizing occurs. When you set it all the way, the bike keeps itself upright very well, and in fact you have to put rather a lot of effort into making it turn. High stability settings are recommended for beginning players so that they can get used to the other controls without having to worry so much about crashing their brains out. This is fairly similar to how it is in real life, you'll notice beginners in motocross don't lean the bike over so far in turns, and neither will you in MXS. As your skill progresses and you feel like you're being held back by the high stability, incrementally turn it down. You'll want to avoid radical changes, because then you'll have to completely readjust your equilibrium all at once, and that's no fun. When I was playing a lot, I toned it down by 5% each day, and that was enough to keep it on the edge of being too difficult, without actually being too difficult to deal with. Eventually you'll reach a point where if you go any lower, you'll crash regardless. At this point, stop turning down the stability slider, and then start experimenting with the advanced stability menu. There's a lot of stuff to tune in order to really get it all dialed in the way you want it, but that should give you the ability to train and develop your skills without killing yourself too often.
Almost everyone rides better in real life than they do in the game at first. Eventually, your game skills will overtake your real life skills, and this can actually help you in real life on the track. Many members on the MXS forums say that playing the game has helped them a lot in real life, particularly in line-selection, body-English, and keeping sharp through the winter months. I have to say, even though I ride trails and dirt/gravel roads mostly, it helped me quite a bit when it came time to turn on the aggressive riding. I wasn't afraid to hang the back end out through a sweeping turn, because I'd done it so much in the game (on a 125, no less) that I knew already what could make it work and what could make it fail.