My 9th Grade History Teacher is teaching that early explorers in the Age of Exploration had a serious worry that the Earth was flat. All the sources I've read online have said it was first discovered by the Greeks and the knowledge of a round world continued through to the time of Columbus. I'm a bit wary of trusting internet sources over a teacher, but the reputation of the validity of this sub is enough for me to trust it.
I haven't personally read accounts of fear of a flat earth, which doesn't mean they don't exist, but this is much more likely to be a later invention. You can read more about what people (including common people) thought about the sphericity of the earth here:
https://m.reddit.com/r/AskHistorians/comments/3d77qn/are_there_any_documents_from_the_15th_century/
https://m.reddit.com/r/AskHistorians/comments/3uufv2/when_did_people_understood_the_concept_of_time/
Tl;dr: people knew the earth was round early on.
As with jschooltiger, I shouldn't like to say that it is categorically impossible that someone may have feared this (after all, I've not evaluated the source material myself on this point). But it seems unlikely in the extreme. As the links bellow note (I've also written about this before on here also), we have no evidence of any educated person after the 6th century arguing that the earth is flat. Furthermore, we have no reason to suppose that it was a popular belief among the uneducated, and at least some tangential evidence to suppose that it was a belief that was taken for granted in things like sermons. Also, for this period in particular, we actually have an extant globe from 1492, finished before Columbus returned from his voyage, which notably doesn't have an american continent in the ocean separating Europe and Asia.
Edit: phrasing.