The slow initial load of your website is likely due to factors like caching, server resources, or network latency rather than the lack of a dedicated server. A shared hosting environment might not perform consistently, especially during high traffic or peak times.
Upgrading to a paid plan can help improve speed, as it often includes more resources, better caching, and optimized server performance. Consider using a Content Delivery Network (CDN) to speed up content delivery worldwide.
Providers like RedSwitches and OVH offer dedicated server plans with higher resources and faster response times, which could further enhance your website's speed and stability.