babek,
There has been so much struggle with the ailing Brave Browser Sync function . . . the first thing for you to try, is allowing, in both devices, ONLY the Sync of Bookmarks, for test purposes.
In addition, make manual backups of the Bookmarks aboard the iOS device; and, make manual backups of the Bookmarks aboard your desktop computer. BEFORE, you test.
When therefore prepared . . . then and only then, remove all the bookmarks from the Brave Browser for iOS, Mobile Bookmarks. Yes, remove all the bookmarks therein.
Because, a prevailing problem, has been, that the Sync
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from the desktop Brave Browser Bookmarks
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to the iOS device Brave Browser Bookmarks
will succeed . . . but not the other way (from iOS device to desktop computer).
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Let us assume, here, that the Sync test, so far, succeeded, and your desktop computer Brave Browser Bookmarks are now populated in the iOS Brave Browser Bookmarks.
Now, create just a single, new bookmark in Brave Browser of your iOS device. Did that new bookmark appear in the Brave Browser Bookmarks of your desktop computer?
If “YES!” . . . then you have Sync working in both directions.
What happens after that, depends upon what you dare, in the form of adding more burden upon the Brave Browser Sync . . . system.
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How to manually backup bookmarks in Brave Browser for iOS, in detail, including several illustrations, step-by-step:
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May help:
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Update - unfortunately:
But “UltraViolet” is optimistic: