Running version is 4.1.
I'm having trouble with one of my blogs. it's rather big and have several thousand entries and tens of thousands of comments.
When I publish an entry, the entry itself gets published almost right away, but the return to the management screen times out and leaves me with a blank page. the url it stops at is this:
http://xxxxxx.xxx/cgi-bin/MT4/mt.cgi?__mode=start_rebuild&old_previous=23659&is_new=0&next=0&old_status=1&type=entry-23661&blog_id=4&entry_id=23661
Also, publishing comments behave the same way. the comment itself appears after a short period of time, but there is no confirmation-screen, and after a while the page times out to a blank page.

Try upgrading to 4.21. I'm serious; there were a lot of performance enhancements to the core in that release, and you also get template module caching which will really decrease the amount of time spent on your rebuilds. It could also be that your shared host is just slow; I used to have that problem when I used Dreamhost.
If you do upgrade to 4.21, I would recommend enabling module caching on both your footer and sidebar modules. The sidebar has A LOT of instructions in it when you count all of the widgets, and chances are, it won't cause any problem for you to have it cached for 30 minutes to 1 hour after each rebuild.
is your blog still slow?
I'm having a similar problem, when I publish a blog that has around 400 entries, it hangs up and stops in the middle of the Entries Publishing. Likewise when I publish the Entry template. I'm left with a blank screen and a long URL. Using 4.31 Open source.
Any ideas?