I don't think the issue lies with the thirtybees system itself; it's likely due to limitations imposed by the hosting provider.
There is a lot of potential here. You can find providers offering managed web hosting at a low cost, though performance and quality vary significantly.
For a standard-sized online shop, it is generally never necessary to use a VPS or a dedicated server. A dedicated server represents a massive security risk that requires management—something you either need to handle yourself (you need to be a sysadmin with extensive knowledge to do this) or hire someone else to do. >> You'd better focus on your business, not on web servers 🙂
Personally, I have had far better experiences using ready-made hosting packages that offer sufficient performance.
Which country are you in, or where do you need/want to host? I’m not a hosting provider myself, of course, but I’d be happy to recommend one or two excellent options where thirtybees runs smoothly with up to 300,000–400,000 products.