I love this monster. It introduced me to the beautiful world of code and let me overcome my fears. For sure, it’s has sloppy and messy code but the more you do, the better and cleaner you start to create. With the only condition: if you want to improve, unless you will continue to design copy-paste-cloned wordpress sites.
Most of the things you can do have been mentioned already, although it seems CDNs weren’t directly mentioned. Many caching plugins will bring that up though, worth a try. Usually only a few bucks for a month.
On another topic, slightly related to the shared hosting point, I recently tried SSD hosting for the first time. It seems blazingly fast tbh, compared to a plethora of other hosting I’ve had experience with.
Worth a try, open one up with an offer of free credit that covers a month or even two depending on the resources you want.
Make a copy of the live site and deploy it there, test directly under the IP you are given. I used Duplicator for this, and a site copy + deploy elsewhere takes minutes.
In other words, no cost besides time spent.