4 data bits suggests 16 level grey-scale. It does indeed sound like a more power chip is needed to drive it - in fact the Parallax Propeller is the one I'd choose with a flexible video-generation unit (well 8 of them...). In fact it is worth googling for prior art with that display.
The arduino should be able to generate the timing signals though, so provide some sort of varying signal to the top data line and get the timing signals right, something should appear on the screen as confirmation.
TFT LCD controllers come in several varieties - ones for generating both gate and source signals for a small display (either video in or frame-buffer), and then the individual gate drivers and source drivers for larger displays (invariable video in).
Here "video in" means some sort of line and frame timing signal plus a clocked data stream.
The single controllers often integrate charge-pump power supply generation (using external capacitors), the one for larger displays will off load that task to a central controller circuit board (partly to ensure equal voltages and partly because of the extra power needed for large area displays).
Some single controllers don't have power supply generation and expect that from an external circuit board (there are various chips that do this task at various power levels)
TFT voltage rails include high and low gate voltages (quite large), high (and perhaps low) "analog" rail(s) for the source-drivers, and logic supply. Of these the analog and logic supplies take most current, gate supplies can be fairly puny (capacitor-diode voltage multiplier). The analog supply feeds a resistor chain to generate the many voltage levels needed for each greyscale or colour level - the outputs of this chain have buffer amps each to provide low-impedance, and these amps are where most of the power goes I think.
The actual LCD gate/source driver chips are long and thin and directly mounted around the display on one of the glass layers.
--- In short its a complex business driving a TFT LCD and different display drivers provide different amounts of support circuitry (such as power rail generation, frame-buffer). Displays for phones tend to integrate most functionality into the gate/source driver (since space is at a premium and the screen is small).