This is only a partial answer.

Given a class $\mathcal C$ of topological spaces, let $S\mathcal C$ and $Q\mathcal C$ be respectively the classes of subspaces and quotient spaces of spaces that belong to the class $\mathcal C$. 

Let $QS\mathbb R$ be the class of quotient spaces of subspaces of $\mathbb R$ and $(QS)^{n+1}\mathbb R:=QS(QS^n\mathbb R)$ for $n\in\mathbb N$. The problem posed in the OP is to describe the class $QS^\omega\mathbb R:=\bigcup_{n\in\mathbb N}QS^n\mathbb R$.

I was able to trace the classes $QS^n\mathbb R$ only for small $n$.

Namely, $QS\mathbb R$ is the class of sequential spaces with countable $k$-network (this can be derived from Michael's Theorem 11.3 in the Gruenhage's survey article in the [Handbook of Set-Theoretic Topology][1]). 

The spaces of the class $SQS\mathbb R$ need not be sequential (but still have a countable $k$-network).

How to describe the class $QSQS\mathbb R$ is a mistery (for me at the moment).

On the other hand, the class $QS^\omega\mathbb R$ is contained in the class of spaces with countable network (which is preserved by taking subapces and continuous images). 

It is a good question to find a space with countable network that does not belong to the class $QS^\omega\mathbb R$.

Maybe $C_p[0,1]$?


  [1]: https://www.elsevier.com/books/handbook-of-set-theoretic-topology/kunen/978-0-444-86580-9