TL:DR: I have InMotion hosting for at least another year. I'm currently on an OLD plan. Despite only having a few sites, they are all slow (7-10 seconds for the initial page). All sites are Wordpress. Is Wordpress hosting actually better than regular hosting? And is it worth the slight monetary upgrade?

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Okay, so a bit more information. I've had InMotion for a few years now (3 years I think) and despite them upgrading their backend a lot, I'm still on an old plan that, while good at the time, seems to be well behind their current plans. At first, my sites were fast, but over the past year and a half or so they've gotten much slower. I've done all the regular troubleshooting using pagespeed and gmetrics, worked with support for both inMotion and my Wordpress theme, but despite this, I can't get a constant decent speed. When I run these, it says the response to the server is the biggest delay. When I explain this to inmotion, they give me 100 things to try that don't work, never taking any blame, telling me it could be my theme while my theme blames them (isn't that how it always is?).   


I'm under contract with InMotion for at least 1 more year at this point and have two options. Upgrade to the new version of my same plan which promises it's faster or upgrade to Wordpress only plan that offers much faster speeds, built-in cache, and more.

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I've been building websites in one way or another for 20 years, but it's always been on regular hosting, so I know little to nothing about Wordpress hosting, so any advice on this (or other ideas) would be really appreciated.  The Wordpress hosting is slightly higher price, but only by a small amount, which will be split between a few clients and me, so the upgrade cost isn't a factor.  


Addition: While 1 site has a good amount of content, I wouldn't consider any of them truly large sites that would bog down a server. Of course, I understand about shared servers and all of that, which is why I originally reached out to InMotion, but they didn't seem to think it has anything to do with their server specifically.
what you want is a form of hosting being sold as "Managed WordPress"

avoid dedicated servers unless you have sufficient ops chops. single points of failure will bite you in the end.