I had this exact same problem from the day I signed up with GoDaddy (about a month ago) until last week. Here’s the short answer: It is NOT your fault, your plugins, or your theme. You need to call GoDaddy support and tell them what is happening. AND you need to tell them you want to switch to a different server. You will have to change servers, to new server, not another old one, or this will keep happening. There’s nothing you can do on your end to fix this.
I worked with their support over a period of several weeks, and talked with a number of good people who may or may not have been totally forthcoming about the problem. I first asked them to add an error log for me (you can do this too, if you like). After seeing that the errors were almost always timeouts, I called again. Another tech told me my server was overloaded with a few sites hogging up all the processing time, that they were aware of the problem, and they were working to move those sites elsewhere. Although I was told the fix would be in “tomorrow”, if anything, things got worse. He suggested I request changing servers at one point, but I was also told I needed to call at least 4 times before they would do that.
Finally, I got a tech who runs a WP site herself, and she told me quite bluntly that I was on an old server (despite being a new customer) that couldn’t handle WP’s demands, that was overloaded, and that they are in the process of replacing. I mentioned the “call 4 times” rule and she said she could initiate a change at any time — and did right then and there. She switched me to a new grid server; it took several hours to complete. And since then my site has been very fast indeed.
So call GoDaddy and let them know you are unhappy. It is, as far as I can tell, the only thing that fixes this problem.