We prove "reasonable'' quantitative bounds for sets in $\mathbb{Z}^2$ avoiding the polynomial corner configuration $(x,y), (x+P(z),y), (x,y+P(z))$, where $P$ is any fixed integer-coefficient polynomial with an integer root of multiplicity $1$. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without $3$-term arithmetic progressions of common difference $z^2-1$. Two ingredients in our proof are a general quantitative concatenation result for multidimensional polynomial progressions and a new degree-lowering argument for box norms.  Joint work with Borys Kuca and James Leng.