You should first have a look at the gas phase spectrum of water to convince yourself that vibrational levels are indeed "discrete". The textbook treatment of vibrating molecules assumes that there is no molecule nearby and each and every unit is independent. Which state of matter is closest to these assumptions? The gas phase, the lower the pressure, the further the molecules are apart (large mean free paths). Therefore if you wish to see quantitization visually, it is better to look at the spectrum of gases (rotations, vibrations and electronic transitions). Recall you study atomic spectrum of hydrogen gas not of solid hydrogen or liquid hydrogen.
Now, you may ask, that water is a triatomic molecules with 3N-6= 3 vibrational modes. What are the rest of those lines? Recall that water can also rotate in the gas phase. Those very thin lines represent rotational energy transitions.
The story changes when you deal with liquids or solids. Now the vibrating molecules are so close to each other that they are no longer independent from each other's influence. The rotational structure is completely smeared. The vibrations are still quantized but now you see a collective behavior millions of close-by water molecules which are strongly interacting by hydrogen bond. If we had a very fancy ultra ultra high resolution infrared spectrometer, we might be able to see some structure in each broad band, but the smearing is so bad in solids and liquids that we all see a overall wide peak, which is just an envelope of all possible vobrational transitions by millions of molecules in different environments on the liquid and solid state.