tl;dr
tr is the wrong tool for the job; try something else (like sed)
tr (text replace) only does 1:1 replacement - so it will only replace one character, with another, at a time. I think your current command replaces > with /n, >> with /n/n, >>> with /n/n/n and >>>> with /n/n/n/n.
try using sed instead, probably something like this (untested!):
cat input | sed $'s/>/\\\n\\\n\\\n\\\n/g' > output