Actually the turbine method is very efficient. The Turgo turbine is up to 90% efficient at extracting the energy from the water. Large-scale hydro plants using Francis Turbines can be as high as 95% efficient (see here).
Electrical generators typically use high RPMs, so you need speed as well as torque to drive it round. Low RPM generators are possible too, but less efficient. Energy available in hydro power comes from the 'head' (height drop) and the 'flow' (amount of water per second). So in low-head situations, you need much more flow to get the same energy. In these cases, you won't get high pressures (and therefore high speeds) without gearing (efficiency losses there too), so a water-wheel approach is feasible. In fact there are low-head generators similar to how you describe - the bucket-and-belt approach - as well as more traditional low-head turbines.
Low-head situations are harder to engineer (economically), because high flows are needed so the whole installation (channels, bearings, generators, etc.) must be much stronger and bigger. It's cheaper (per kW of generating capability) to use high-head situations with smaller/faster turbine wheels at higher pressures.
So these are the main reasons you see mostly turbines instead of other types of hydro plant. As technology improves and low-head locations become cheaper to exploit, we will see more of them.
