Why invading Italy made more sense than liberating Norway

Would the Allies have been better off liberating Norway instead of invading Italy?

No. Norway was even harder to sustain, while the Italian campaign diverted German forces and resources from the coming invasion of France.

Norway's mountains, narrow routes, and long travel times make offensive logistics exceptionally difficult even after a landing succeeds. Italy was also hard terrain, but fighting there forced Germany to commit resources that could not reinforce France. Germany's surprising 1940 success in Norway is poor evidence that an Allied liberation campaign would have been easy.