I’ve run into issues before with having too many post revisions stored in the database. Cleaning them out helped. You can run this query to see where you’re at.
SELECT COUNT(*) FROM wp_posts WHERE post_type = "revision"
if you have wp cli installed it’s as easy as
/path/to/wordpress/ $ wp db query 'SELECT COUNT(*) FROM wp_posts WHERE post_type = "revision"'
If that number seems high (I can’t really say what high is, depends on how many posts you have) you could make a backup of your db (!), delete all the revisions and see if that speeds things up.
/path/to/wordpress/ $ wb db export /path/to/your/backup/home/mysite.snapshot.sql
/path/to/wordpress/ $ wp db query 'DELETE FROM wp_posts WHERE post_type = "revision"'
There are plugins to make deleting a little more nuanced. Say, delete all but the last 2 revisions or whatever.
If that turns out to be the problem, you can permanently cap the revisions count in your wp-config with this line:
define( 'WP_POST_REVISIONS', 3 );
More info here: http://codex.wordpress.org/Editing_wp-config.php#Specify_the_Number_of_Post_Revisions
