Riemannian geometry texts note that the Christoffel symbols, sometimes written: Γijk, do not transform as tensors. On the other hand an affine connection on an affine manifold is a tensor. What gives?

The Christofel symbol can be thought of as describing how a coordinate system twists and stretches as you move about on the manifold. It thus is about a particular coordinate system in two different ways:

The affine connection, on the other hand, speaks of no particular coordinate system but is, as any tensor representation, expressed in some coordinate system.

It is possible to express the twisting and stretching of one coordinate system in another coordinate system. If we keep the latter coordinate system unchanged then that is a tensor expression referring to some subject coordinate system.

This is the source, I think, of the confusion in this discussion of the covariant derivative.