Our experience is different with SiteGround, and I'm not sure I can recommend it the same way it is recommend it by others. In fact, we don't recommend any of the mainstream hosting providers to our clients (those with "gator", "monster", "ground", "daddy", "temple", and "cloud" in their domain), simply because they have the small client in mind and so: 1) they have all these draconian firewall rules, 2) they have aggressive caching methods, 3) it is often hard to get a real support person, and 4) they don't take the client's business seriously. (Note: I can easily elaborate on all of these 4 points.)
It is a good idea to search for a host which does not serve small clients, and only caters for medium to large clients. It is also a good idea to stick with a host in North America (avoid all these hosts that are based in Europe or Asia). Finally (for this subject), it is important that you choose a WHM based environment (make sure you have the root password). I have no idea why there are many servers using Plesk/Webmin/Other, which are a huge pain to work with.
Now, going back to your original question, unless I am very mistaken, you will not be able to get this working with a shared hosting package. 500,000 /month is around 15,000/day, which is a lot, for a Joomla website. You will need a dedicated server, with at least 16GB of RAM. You will also need to use InnoDB for your Joomla database as MyISAM is unreliable and slow (by the way, MyISAM used to be faster than InnoDB, but now it is no longer the case). Now, once your website is up, you will need to constantly monitor your slow query log and optimize any queries with the problem (optimize the core when you must). Always ensure that your load average is less than 2. Ideally, it should be less than 1.
If, by any chance, you get your Joomla website to work on a shared host, then you are likely paying for a shared host more than you are paying for a dedicated server, or it might be that your host is using some aggressive caching tactics that will make the management of your Joomla website a complete mess and an experience in misery.