Shoot down x Turn down: Qual a diferença

I think perhaps "shoot down" was used metaphorically, as in (compared to) shooting down an aircraft in flight. And if spoken aloud in the street, depending on where someone is, could be misunderstood as one having aimed and shoot someone else with a gun.
Shoot down could be also "rejected" as some legislation having not passed.

Other than that, I assume that it was used in a slangy way.

Turn down é muito mais comum e é a opção mais adequada, nesse contexto que você mencionou.

Tenha cautela ao usar shoot down, pois este phrasal possui outros significados.