Gimp will allow you to create bezier curves when editing, but as soon as you add a stroke or fill to the curve the result will be a raster image.
InkScape is IMO the best free alternative to AI. (Why do you hate it?)
BTW if you go for inkScape, use the optimized SVG option, to remove unnecessary metadata, comments etc.
inkscape.JPG 38.31KB 0 downloads