Drupal messages won't disappear
Messages are kept in the user's session, which by default is stored in a database table called 'sessions'. If your database server dies partway through the page request, the messages may not get cleared from the session.

You can see the MySQL server failing when updating the sessions table in this image due to a max_allowed_packet error.

1M is a very small max_allowed_packet for a Drupal installation. You may need to ask your host to increase this limit. Alternatively, you could turn off dblog and use syslog instead, which sends your Drupal logs to a system daemon - but this may require your host to set up a syslog location for you. 

There may be a module that implements hook_watchdog() that can save logs to a standard file. I can't find one, but if one exists, that could be a good option in your situation too.