Remove the noise first. Whats your hangup with that?
Then compress over your threshold and add make up gain.
Depending on the noise and level you may not be able to achieve what you want without a better source recording.
Normalization raises everything so the peaks are at -1 in audacity. You can choose another peak if you dont like the default.
The trouble with Normalisation is it is undiscerning, it simply increases all gain until one peak reaches 0dBFS - & brings the entire noise-floor with it by the same amount.
Adding compression is only going to make this worse, as it's limiting the peaks, allowing the noise floor to come up further.
The simplest method is going to be to use the '...