Most web hosts can run Joomla sites well, some better than others.

5 tips to help you in hosting your Joomla site.

1. Use Unix or Linux servers.
Joomla was designed and built to run in a unix/ linux environment and it performs the best here.  Joomla will run on windows but not as well, you are more likely to have permission, performance, and php issues when using windows.  Avoid using windows if possible.

2. Cpanel - use a host that has cpanel for the control panel.
There are many good control panels available.  Joomla seems to have fewer issues when run on cPanel servers.  In addition to fewer issues it will be easier for a Joomla professional to diagnose any problems you may be having.

3. Try to use one site per control panel
This can not always be done but it is better to use one control panel or hosting account per joomla site.  True this does become more expensive but it will be easier to manage.  Additonally if some one is able to hack one of youre websites you are able to stop the damage.  If you have one account that runs 30 sites and it gets hacked all 30 sites are at potential risk.  If you only have one site in the account and it gets hacked then you only have one site that needs to be cleaned up.

4. Get and Uptime Monitor
All servers go down and run slow from time to time.  A good uptime monitor will keep you informed about your site or sites.  Also if your hosting server does start to have an issue you can see the performance over time and present the information to your hosting company if needed.  An uptime monitor is and addtional expense but well worth it.

5.  If someone is assisting you with your Joomla site go with there suggestion for a host.
If the hosting of a site is a little too complicated and you are getting help, listen to who is helping you.  It will be much better to go with a hosting company that they are familiar with than forcing them to start over new.  If they are helping you, make it as easy as possible for them.  Then they willl be more likely to keep assisting you.