Other problem seen (I suppose they are related) : on this server, the MySQL
process is sometimes consuming a lot of CPU, and we need to restart the server.
I'm guessing this is more the issue. You'll need to study up on tuning MySQL server to perform better or hire an expert in the field to do it.
I googled, but can't find any solution to these problems. I just realized that the PHP memory limit is only set to 48 MB, which is a small value I think. I asked the provider to increase it to 128 MB.
Once you begin upgrading to Drupal 7 it is a good idea to increase the memory from 48 MB. You'll need to look at your hosting service plan to determine how much to give each process.
Hi ! Sorry for my late answer, I worked on other projects the last days.
I had a look to the server, and realized that it was running 9 websites : 6 Drupal websites, and 3 Joomla. This server only have 1 GBytes of memory...
I think it's not necessary to try to understand the problem anymore!
Thanks for your answers! Fred