Just a bit more to add -- I think this is a really useful visualization to supplement the explanation above. It's mathy, but a good combination of physics and road safety. Here I'm showing the maximum acceleration we can get by braking before we lose control, for any given curve and any given starting speed through that curve. I assume a coefficient of friction of [math] and [math], and I ignore the slope of the road since the effect is generally small. This is a function of two variables, so the graph is actually a surface in three dimensions. The vertical axis is our max acceleration in meters per second squared. The R axis is the radius of curvature (in meters), and the v axis is our speed (in kilometers per hour).
And the function which makes this graph is[math]
which is derived by calculating the combined acceleration (centripetal plus linear) with acceleration being due to friction.A few bits of insight to be had here. First is that for low speeds and large radius of curvature (gentle curves; large values of R), the function is pretty much flat and near its maximum. The maximum is given by [math]. In other words we can achieve at most about 70% of g. Sporty vehicles with specialized tires can do better, while poor road conditions make it worse. (No surprise that it's very hard to start or stop on an icy road).
Next, as the turn becomes sharper (smaller R) and our speed becomes faster, our maximum safe acceleration decreases. But it does not decrease in a linear way. It's very gradual at first, but then plummets very steeply. That steep region is what we want to avoid -- it means we have less control, and the amount of control we have changes very quickly. A few km/hr makes a big difference. Beyond that region, the graph drops to 0. Actually, the function is undefined here, and represents combinations of speed and curvature for which we cannot be in control.
For more clarity, here's the projection in the v,R plane. I think this really emphasizes the 'risky' area where we are close to that maximum speed calculated earlier.
Of course, this is all just numbers and nobody is seriously going to do math to figure out if they are driving safely. The goal is to understand conceptually why we can lose control on curves even when we think we're doing the right thing, and what we can do to help mitigate the risk.