"bright moon" is correct and usable in written English.
You could use it to describe the night sky, for example: "On a clear night, the sky was illuminated by a bright moon."
And that's when we walked straight into a bright moon.
A fine bright moon and thousands of stars!
A bright moon will diminish the viewing somewhat.
He stood there under a bright moon, with his finger to his lip.
The New Yorker, November 21, 1977 P. 58 A fine bright moon and thousands of stars!
A bright moon can astonish, no matter how many times you have seen it.
Past the Chrysler mast bright moons have gone.
A poem from the Tang dynasty describes a service of teacups for the emperor as "bright moons cunningly carved and dyed with spring water".
The most distinctive works here combine velvety watercolors of bright moons and silvery clouds with more obdurate things -- a ladder, slate shingles, an ancient bird cage filled with little bundles of sticks.
Then, in 1610, he burst onto the international stage with "The Starry Messenger," announcing his discovery, via telescope, of the craters and mountains on the moon, the four bright moons of Jupiter and the fact that the Milky Way is composed of millions of distinct stars.
The Earth will move between the sun and the moon, so that a shadow will be entirely cast over it — sending the previously super-bright moon into darkness.
