Welcome to the Q&A Forum

Good discussion going on here. I did want to highlight this Google Forum discussion in case it would help with this topic: https://productforums.google.com/forum/#!searchin/business/language/business/m9ye2hHR5Nw/nlzE_WQgKf4J

I have some experience in this as all of our websites are in at least 2 languages - some in more.  I have found that I have had to do SEO seperately for each translated version of the site to get them to rank.  So, for the Spanish version, I had to work on creating links back from Spanish sources specifically to the Spanish pages.  For the Russian pages, Portuguese, Japanese, Chinese and Korean pages I had to do the same.  The SEO work I did on the translated pages did not seem to have any noticeable benefit to the other languages SERP even though the pages are all like together. It sounds to me that since you guys are primarily a French site, it may be that you got most of your links back to your French pages, from French sites.  If that is the case you may try to work on specifically getting links for the English section. I hope this helps.