MPEG-4 is a video steam type that the .avi container uses.
.avi is the container - the extension has nothing to do with crappiness.
The codecs and their settings that Final Cut is using determines the quality of the streams within the resulting container, not the choice of container. As the software is a branded product, they likely dumbed down the codec settings used when exporting to .avi
First, have a poke through settings and see if you can choose the codecs used in conversion. If you can, set .avi to encode video using XviD, mpeg-4 or h264 @ ~1000 kbps 25fps and audio at 48000 ~128kbs cbr.
If you can't setup the exporting codec settings, dump the file raw or export in a DVD format (MPEG2) and use a real encoding program to make the conversion to .avi (or even directly to .flv).
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