The Raman frequency of the intersubband charge-density excitation plasmon in type-I and type-II quantum wells is known to be very sensitively controlled via a low-power optical pump signal. We find that, above a threshold electron density of approximately $4\ifmmode\pm\else\textpm\fi{}2\ifmmode\times\else\texttimes\fi{}{10}^{10}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}2}$, the charge density plasmon mode splits first into two, and then with increasing density into three closely spaced frequencies. A similar splitting occurs in the spin-density wave plasmon associated with the same intersubband transition. We analyze the results including the coupling to the longitudinal optical phonon and hypothesize that the splittings arise from the special situation of a structure of spatially separated bipolar plasmons.