There are multiple different designs, or wildly varying complexity, so to answer your question in one phrase: it varies.
In the most common case there's no gas generator as such. Fuel and oxidizer are injected into the combustion chamber from pressurized tanks, ignited, and the burning process takes care of both turning them from liquid to gas, and then of the propulsive combustion. No gas generator, no turbopumps, just some regulated valves, an igniter and the combustion chamber with nozzle.
Then there are designs of increasing complexity:
- channeling the fuel and/or the oxidizer through the bell nozzle, both cooling it and turning the two into gas for better mixing.
- driving the turbopumps with an entirely separate propellant, like hydrazine.
- driving some exhaust gas off the combustion chamber to propel the turbopumps and/or evaporate the cryofuels.
- using the channeled, heated gasses to propel a turbopump that pumps cryo-fuels. (which is pretty much the case that you doubted. Actually, quite viable as the fuel/oxidizer after boiling off in the walls of combustion chamber has much more energy, and is well capable of driving the turbopumps.)
- using pre-burners, where fuel and oxidizer are mixed in very different than optimal ratio, both evaporating the liquids (for more smooth combustion) and providing energy to drive turbopumps.
- using separate gas generators, where hot combustion products (be it from the nozzle or preburners) boils the cryofuels, and separate pre-burners that provide heat and energy for the turbopumps.
- ...and any wild combination of the above.
This pretty much culminates with the design of the Space Shuttle Main Engine. The public-domain presentation, page 19, shows its gory details: four turbopumps (each with distinct turbines for propelling them and for pumping), two preburners, a gas generator that evaporates LOX, a bell nozzle cooling grid that boils off LH2, and a byzantine grid of connections that drive the fuel and oxidizer heated, hot partially burned oxygen-rich, cool but evaporated. re-cooled after passing the turbopump, heated but not evaporated after being pushed through turbopump, superheated through pre-burning fuel-rich, re-cooled by passing through another turbopump, injected back into tanks to provide pressure, back and forth through the grid, in a way that makes sense if you really put your mind to it, but is hard on one's sanity.
So... your understanding is incomplete ;) The bottom line is there's a lot of designs. The one in the picture you linked is pretty straightforward: some propellant and oxidizer diverted from the main lines gets burned in a preburner (the "gas generator") providing work for propelling the turbine. It may or may not be used to ignite the proper fuel/oxidizer mix. It may or may not be used to heat the oxidizer. The schematics depicts a generic, pretty straightforward, example middle-range rocket engine, and the exact details in reality may vary wildly.