To avoid this issue, try changing the interlace mode of the conversion. You can find it in the Settings menu for the Video options in Handbrake. You will see that every mode used in the Deinterlace menu offers different image quality. The video doesn't matter, only the file format. The deinterlace settings work different for certain video file formats.
If these settings do not fix the issues, then I recommend checking the tutorials available through the Google Search. It's possible that you're missing settings, when the conversion takes place.