There are two types of normalization: peak and RMS. Audition performs peak normalization, and this doesn't impact the dynamic range. RMS normalization will change the dynamic range, though I forget which apps support this feature.
As you note, stan, plain old amplification raises the level of the entire waveform by the same amount, thereby preserving dynamic range. What you're looking to do is decrease the dynamic range by lowering peaks that rise well above the rest of the music, so you may then raise the overall levels without those peaks overloading. Several options:
<1> hard limit -- quick & dirty, doesn't sound as good as the other options, but it can be easy / useful on transients
<2> compression -- a softer way of reducing peaks, and therefore dynamic range, takes a bit more work to find the right settings depending on the source
<3> volume envelope -- the most flexible of the options, but more effort than <1> and <2>, depending on the source
Alternatively, you could use one or more options at different spots in the source WAV. If there are a lot of peaks I want to lower throughout the source WAV, I'll often apply compression to the entire waveform, and adjust the threshold and ratio to get the right mix of compression. If I only need to adjust a small number of peaks, or need to apply variable settings to achieve the results I like - e.g. using X settings on peak A to achieve the results I want, while using Y settings on peak B to achieve the results I want - I'll apply one of thet three options on a spot basis. Hard limiting is quick & dirty, and doesn't sound as good; compression and volume envelope sound better, but take a bit more time to sort out the best settings.
In Audition, main menu access to the above options: hard limit = Effects | Amplitude | Hard Limiting, compression = Effects | Amplitude | Dynamics Processing, and volume envelope = Effects | Amplitude | Envelope. I find Audition's envelope function difficult to use - one manipulates the envelope in a pop-up dialog window, instead of in the main waveform view, making it difficult to map specific changes in the envelope to specific sections of the waveform.