Python extension modules written in C are faster than the equivalent programs written in pure Python. How do these extension modules compare (speed wise) to programs written in pure C? Are programs written in pure C even faster than the equivalent Python extension module?
How do these extension modules compare (speed wise) to programs written in pure C?

They are slightly slower due to the translation between Python data structures -> C types. Disregarding this translation the actual C code runs at exactly the same speed as a regular C function would.

Are programs written in pure C even faster than the equivalent Python extension module?

C programs (written entirely in C) can be faster than Python programs using the C extension modules. If the C program and the extension module are written with the same level of complexity, coder skill, algorithmic complexity, etc., the C program will win every time. However, if you're not a C guru and you're competing with a highly optimized Python C extension Python could be faster.
Speed of Python Extensions in C vs. C