This is a hard problem. There are reasonable approximations like Tiller-Hanson (see my answer to this question: How to get the outline of a stroke?) but the questioner specifically raises the difficulty that 'the normals can cross each other in steep curves'; another way of looking at it is that an envelope created using normals can produce an indefinitely large number of loops, depending on how closely spaced the normals are.
A perfect solution, without self-intersections, is the envelope of the Minkowski sum of a circle and the line. I think it's impractical to get such an envelope, though: you may have to accept the intersections.
Another interesting but daunting fact is that, as Richard Kinch notes in MetaFog: Converting METAFONT Shapes to Contours, "Algebra tells us
that stroking a 3rd degree polynomial curve (the ellipse
approximated by Bézier curves) along a 3rd
degree polynomial curve (the Bézier curve of the
stroked path) results in a 6th degree envelope curve.
We will have to approximate these 6th degree exact
envelope curves with 3rd degree (Bezier) curves".