I'm not familiar with your growing conditions but I think you're looking for "understory" trees and they do exist (dogwood maybe?). The "growing quickly" bit might push it into unicorn territory though. :)

2 suggestions: Gray dogwood, or Arrowwood viburnum. Fast growing (especially if there is adequate soil moisture), shade tolerant. The dogwood can be pruned to either small tree form, or left as a thicket of shrubbery. The arrowwood I find harder to make into a tree -- every time I cut it back, it sends up tall, straight (arrow-straight?) branches. But it's pretty, and certainly tall and thick enough for privacy. Both have berries that birds enjoy.

A bit slower but would be fun, is a pawpaw tree.