use Share/Export Using Quicktime
in its Options try …
- h.264 as codec
- reduce resolution (1280x729, 640x360)
- a lower frame rate (15 instead of 30)
- latest attemp: lower bitrate
… should result in a smaller file. Quality is most of all determined by the original material! Your original recording is probably highly compressed and optimized for playback; after edting, processing and export-encoding, any 'flaws' in the original get multiplied.
Don't expect wonders with high compressions