No no no! Big O has nothing to do with Turing machines at all! You can, of course, express the number of steps a single-tape deterministic Turing machine needs to solve some problem using Big O notation. But you can just as well express the number of Qubits that a quantum computer needs to solve some other problem in Big O notation. It has nothing to do with the quantity being measured. Are you perhaps mixing up the definition of complexity classes such as P and NP with Big O notation?

First you need to settle on a model. Then you can express time, space, or some other quantity in that model using Big O notation.

Arguably, I've found that HN is one of the worst places to be precise about Big O notation. Everyone here seems to think they know what it is, only to lay bare their misconceptions about it a second later.