However, it has a lot to do with how the hosting is set up.
Weibo, which started out as China's version of Twitter, basically, was serving millions of people with just one four core server for awhile, as their Twitter clone was written in node.js at the time. That's super light on the server, transferring most of the execution to the clients (the users). Very smart! They obviously got FAR more out of a tiny server than anyone thought possible.
That said, it's often specialty hosting you want. This is one of the few cases where a specialist ends up being both far better AND cheaper.
I've been looking very hard for PrestaShop only hosts, much like you can find Magento and Wordpress specialist hosts. Those are often the key to getting the very best performance out of a script without YOU being the one extracting that performance. Specialists know how to set up all the many layers just right.
I've used both specialty $100 Magento hosting, and now $5/mo specialty Wordpress hosting, and both perform far better than any host with a generalist setup. Even a dedicated server with 12 to 48 cores isn't going to beat a specialist host that actually sets things up properly.
Unbelievably, I can't find a PrestaShop only host! That's the thing that blows my mind.
Without this, before switching servers, I'd say switch over from Apache or nginx over to Litespeed server first. The host I use that can accept PrestaShop AND has Litespeed is GeekStorage in Chicago, but there could be ones that are better for you in your area.
There's also a little add-on here which allows you to use LIteSpeed's native caching. LiteSpeed has hole punch caching, which is very advanced (in other words, it can simply cache the parts of the page which never change). Really quite brilliant.
LiteSpeed can also absorb smaller DDOS attacks. It's so good, some hosts fit something like 10 or 20 TIMES more users on a server with LiteSpeed.
So I'd look for that first. A $200 to $500 server, setup properly, will probably work just fine if you're running an optimized LiteSpeed and a heavily optimized MySQL setup. There's alternatives to plain MySQL, too, such as Persona, which is a really good host can setup.