I use bluehost, overall I'm happy enough. Do you do anything to optimize your site at all?
How many plugins do you use? Is your theme lightweight? Do you cache at all?
These are all things that affect your site's load times. Also, what do you mean by 'bigger site'?
Depending how large, you'll find that no shared web hosting will work. If you mean big in terms of a lot of content, that's fine... but if you mean big in terms of traffic, you'll outgrow your hosting. I can handle a decent amount of traffic on bluehost with caching, cdn, and cloudflare.
You can see how much you might be outgrowing BH in your cpanel, there is a little icon to check on throttling. If your site hogs too many resources, you get throttled. Check the chart out and see how often they are throttling you. If you are being throttled, you can remedy it using some of the options I mentioned above.... but like I said, if you get too big, no shared/web hosting package will work. You'll need to move on to soemthing bigger