It's hard to say exactly what the problem with the given information but with what you have lead to say its most likely a hosting problem.
Shared Resources and Fixed Limits
The problem with shared hosting is that various web host providers will run there shared hosting differently, it sounds to me that site5 has set max resources on their shared accounts meaning your only assigned a certain amount of CPU time meaning that it will slow down on medium to heavy tasks.
Take GoDaddy as an Example
The reason I say this is that I have many shared accounts off over the globe with about 12 different shared hosting companies, naming one of them GoDaddy I am assigned fixed resources and my site really slows down if I purge some of the SQL - in fact all of the other GoDaddy hosted sites slow down at the same time which proves that they are restricting the amount of resources I have and even in there terms it says 'UNLIMITED SITES*' which mentions that each site I add will have the same shared resources so effectively the more sites I add the slower it gets as the usage goes higher, however on other shared hosting accounts this isn't the case and they just share the whole resources across every website on it, this means you get burst but also means that if some sites are heavy you effectively can be slowed down because of them so its a win lose situation.
Consider VPS Hosting
VPS servers are really cheap these days and much faster than shared hosting, in some cases VPS can be cheaper, Take OpenVZ for example these tend to be the cheapest.
I recommend you take a look at VPS Clouds, Xen, OpenVZ, and KVM and pick one you feel is in your budget and requirements.
Once you go VPS you won't go back.