The exact same thing happened to me. I outplayed my opponent who had a winning position, and what should have been a good-natured handshake at the end, was accusations and suspicions. It was stupid. What a waste.
I actually had to end up blocking him because I was worried that if we played again, he would feel free to "prove" his suspicions. There is no good that comes of bringing up the topic. None at all.
Thanks to the echo-chamber in the forums, created by people that participate in speaking about the matter carelessly, there is the perception of an epidemic that simply does not exist. Period, point blank.
You asked, "Are there so many cheaters?" No. There aren't. They're almost non-existent.
Is there that much unfounded suspicion? Yes. There absolutely is.
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1. There is hardly any cheating in online chess.
2. Lichess handles it the best.
3. If you suspect someone of cheating report them.
It's as easy as 1-2-3. Anything else is counter-productive and harmful.
Other than that, it's time to eviscerate this pervasive echo-chamber that does absolutely no good and only causes bad upon bad upon bad.
It is rotten and needlessly spoils everything.
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This explains my point perfectly:
www.youtube.com/watch?v=X1Wiok4H1RY