The short answer is pressure. What you've described above is basically just osmosis, and without applied pressure to overcome the osmotic pressure of the system, reverse osmosis doesn't happen.
This Wikipedia article gives a good description of the concept, and the difference between filtration, osmosis and reverse osmosis:
Osmosis is a natural process. When two solutions with different
concentrations of a solute are separated by a semipermeable membrane,
the solvent has a tendency to move from low to high solute
concentrations for chemical potential equilibration.
Formally, reverse osmosis is the process of forcing a solvent from a
region of high solute concentration through a semipermeable membrane
to a region of low solute concentration by applying a pressure in
excess of the osmotic pressure. The largest and most important
application of reverse osmosis is the separation of pure water from
seawater and brackish waters; seawater or brackish water is
pressurized against one surface of the membrane, causing transport of
salt-depleted water across the membrane and emergence of potable
drinking water from the low-pressure side.
The membranes used for reverse osmosis have a dense layer in the
polymer matrix—either the skin of an asymmetric membrane or an
interfacially polymerized layer within a thin-film-composite
membrane—where the separation occurs. In most cases, the membrane is
designed to allow only water to pass through this dense layer, while
preventing the passage of solutes (such as salt ions). This process
requires that a high pressure be exerted on the high concentration
side of the membrane, usually 2–17 bar (30–250 psi) for fresh and
brackish water, and 40–82 bar (600–1200 psi) for seawater, which has
around 27 bar (390 psi)[6] natural osmotic pressure that must be
overcome.