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I'm a bit curious, it sounds like you haven't redirected the old site to the new one? If you have, have you performed a change of address in Google Search Console? And lastly, are there language differences between your .fr and .com sites? Have you set up your hreflang tags on one and not the other? All else being equal, if you have two sites and one is optimized for a french audience via hreflang tags and a .fr domain, it would not shock me at all to see that Google is displaying one over the other on google.fr. Also consider that your french users could be clicking on the .fr site and boosting its ranking that way, causing Google to consider your old domain a more relevant source of information for your audience. I very much doubt your competitor is affecting your ranking. If anyone could bowl sites on Google with that kind of cheap tactic they wouldn't have a working search engine in today's day and age. More likely your competitor is harming his own rankings. Switching hosts will do nothing. Stick with GoDaddy and save yourself the trouble and expense.