First check that you meet the minimum requirements, especially as far as the PHP version is concerned. Make sure you have at least PHP 5.3.10 (recommended: PHP 7.2.17). Oftentimes the reason you don't see an update is that your server simply does not meet the minimum requirements.
If you are sure your server supports Joomla! 3.9.6, the actually latest release at the time of this writing and not 3.8.3 as you mistakenly said, first take a backup of your site. Then follow Method C per the Joomla! documentation.
If I recall correctly, your Joomla version also includes the Upload & Update feature I contributed to the Joomla! Update component. If this is the case we can follow Method D, not mentioned in the documentation:
- Take a backup of your site. If things go wrong this is your only way to restore your site back to normal.
- Download the latest update package from 3.x (e.g Joomla! 3.x to 3.9.6)
- Go to Components, Joomla! Update, Upload & Update
- Choose the ZIP file you just downloaded and click on Upload & Install
- It will ask you to re-enter your Super User credentials. Please do that, it's a security check.
- Joomla! Update will now use the ZIP file to update your site to the latest version.