LR 5 does take more horsepower to drive its UI than LR3. The Macbook air is set up more for battery life than raw performance, so it's not as powerful as a macbook pro. Still, it shouldn't be that bad.
A few things to try:
optimize the catalog.
if you're running with a second screen enabled, turn it off. I found that really slowed Lightroom down on my older laptop.
select all your images, the choose Library-Previews->discard 1:1 previews, then
choose Library->Previews->Build 1:1 previews (then go get some coffee).
It's possible your previews are poor, and it's regenerating them every time you switch images. Building up a set of previews will take time, but speed it up.
Note that if you convert things to DNG the conversion process can be quite slow.
Also consider building a full set of Smart Previews.
In Preferences->File Handling check the size of the "Raw Cache" setting. If it's too small, you'll be throwing out and rebuilding your image caches too often. I set mine to 20Gb. Also I'd suggest purging it in case something's messed up in the cache. Ditto the video cache below it.
In "Catalog Settings->file handling" you probably want to adjsut your preview numbers. Mine are set to 2880 and "high" and discard 1:1 previews to never. For that computer, I'd go with 1440 and medium. Discard 1:1 previews probably should be at least 1 week, or you'll be throwing out and regenerating previews a lot.
See if any of these help.