"My child is so smart."
Translation:Moje dziecko jest takie mądre.
"Moje dziecko jest takie bystre." is another perfectly correct solution. I'm an educated native speaker of Polish.
Does "mądre" mean "well dressed" (as in British English), or "intelligent" (as in American English), or both?
No, mądra/mądry/mądre does not refer to any other thing as quality of mind or sth that was done according to it. The simplest translation is wisdom. One can „Mądrze się ubrać” which means that one took into account some factors like temperature or rain.
It's not Americanism. It is only opinion if something was done reasonable or not. That's all.