Look at your many options (and there are many!) ... but also make sure your code is as fast and lean as possible.

Moving to the next level is based on ROI ... if it is not there don't bother ... else it becomes a money pit.

Remember that the costs of moving to a dedicated server are more than just the monthly price difference. A dedicated server means YOU are responsible for updates/patches to the OS, database, application language (PHP, etc.), firewall, etc. Do you (or someone on your staff) have the technical skills to handle that? If not, you'll need to hire a consultant.

Another option is the middle ground of a managed server. It's a standalone server, but the hosting company is (supposed to be) responsible for updates/patches.

Second the managed server ... there is some benefit in overall site performance but whether is becomes a speed demon or not I cannot say. Shared has some amazingly good times if managed properly. Managed, however, does take out some of the "tech" stuff and you can call 'em up when they fall short and get things fixed.

Speed is many things ... and most of them are on YOU, the webmaster, not the iron you are running on.

Dedicated is best if you know how. Virtual servers are still shared allotments on a server, sharing memory CPU and bandwidth which has limits per server and connection. So depending on how many other virtuals are running and whether they are busy or not, your performance can waver and at worst come to a standstill with timeouts.

Using a reputable host is best and one thing is for sure... what is cheap is not good.

As in all cases, vet the host!

Some are very good, others not so much.

An underutilized shared server can give better performance than a mediocre dedicated server. The amount of websites served by a server is no indication of the relative performance of each of them. If these sites are serving one page a day, I would be happy to have thousand on my server.

What is relevant is speed, and you are asking for recent customer engagement metrics. I recently changed a site with heavy scripting from PHP 5 to PHP 7 and moved to HTTP/2 serving. The PHP move reduced the execution time of the scripts by a factor of two because PHP 7 is more optimized, and the HTTP/2 serving made loading image-heavy pages more snappy. The bounce rate dropped from 40% to 32% due to these changes. If you move to your own server you have the choice to select the versions of programs and protocols you want. But you also need the knowledge to do it which comes at a price.

If your site is dynamic and uses a database, another important question is where the data is located. At the same server, or at a separate database server. On a number of shared setups the databases are located at a separate server. This can be good if the database server is beefy and underutilized, but it can be bad if the network path to the database server is congested or other sites are using the database server heavily. Moving all to a dedicated server may have a positive or negative performance impact depending on this. Again, your technical knowledge to implement a database server (as a process or separate server) comes at a price.

Thanks for all your responses guys! It's good to have some other opinions :)