Remove the noise first. 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 don't like the default.
You could try to remove noise a second time before compressing and adding make up gain. Not sure you could achieve much with that but worth a try once.
