Bézier curve in old school racing game

You might want to consider using a Catmull-Rom spline instead - I find them easier to fine-tune because every control point lies on the curve.

Edit:

But if you want to stick with Bezier curves, you'll want to create a piecewise quadratic or cubic Bezier curve with C1 continuity (i.e. the end points of connected curves meet at a common point and share a common tangent). See below for an approximation of the left edge of the road using two cubic Bezier curves.

There's a cheap yet effective way to draw beziers... if you have access to triangle rasterization and shading (it's unclear to me if you do).
Use bezier triangles (optionally inverted) and put their three vertices on the red spots you marked. The technique is used in Resolution Independent Curve Rendering using Programmable Graphics Hardware.

Another way - which also requires some kind of HW acceleration is Green's method based on distance maps. Improved Alpha-Tested Magni?cation for Vector Textures and Special Effects.

The latter is more flexible IMHO and comes at an extremely reduced effort although it delivers slightly less quality (it's still very good). I only suggest the first for "true" vector renderers.
If there's no access to 3D acceleration, I'm afraid you will have to write everything yourself. Hopefully this is still not so difficult given those two limited tests.

Thank you guys. I think (not sure) everything is software rendered because I use a canvas.getContext('2d'). [color=#282828][font=helvetica, arial, verdana, tahoma, sans-serif]

The Catmull-Rom spline seems to be a good solution.

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