Ill always take an opportunity to say CiviCRM can and will work just fine with shared hosts! However not all shared hosts are created equal. Two that support CiviCRM requirements include:
- CiviHosting - Supporting Partner of CiviCRM
- Namecheap (Stellar Business) - Not a supporter of CiviCRM.
Each of these hosts have met my needs on a variety of sites. GoDaddy I believe uses CPanel - in CPanel - choose "Select PHP Version" - make sure the right version of PHP is selected (should be 7.1), if not, make sure you select the correct version as current. Then make sure all the options needed are selected. Finally switch to php options and make sure all those settings are correct too.
The requirements for CiviCRM can be found here:
https://docs.civicrm.org/sysadmin/en/latest/requirements/
I should add, shared hosting does have its limits - depending on your needs, your organization may need something more substantial
Updated 03/22/2019: Removed TMDhosting as they are no longer a supporting partner, removed stablehost, as I can no longer recommend their services.