  Let $N$ be a large prime and $P, Q \in \mathbb{Z}[x]$ two linearly
independent polynomials with $P(0) = Q(0) = 0$. We show that if a subset $A$ of
$\mathbb{Z}/N\mathbb{Z}$ lacks a progression of the form $(x, x + P(y), x +
Q(y), x + P(y) + Q(y))$, then $$|A| \le
O\left(\frac{N}{\log_{(O(1))}(N)}\right)$$ where $\log_{C}(N)$ is an iterated
logarithm of order $C$ (e.g., $\log_{2}(N) = \log\log(N)$). To establish this
bound, we adapt Peluse's (2018) degree lowering argument to the quadratic
Fourier analysis setting to obtain quantitative bounds on the true complexity
of the above progression. Our method also shows that for a large class of
polynomial progressions, if one can establish polynomial-type bounds on the
true complexity of those progressions, then one can establish polynomial-type
bounds on Szemer\'edi's theorem for that type of polynomial progression.
