Yesterday I turned off "
Use software rendering instead of GPU rendering" in Internet Tools, Advanced tab (as suggested by Dreamer here...
http://forum.avantbrowser.com/viewtopic.php?f=6&t=30686#p180163 ) and it does not seem to have stopped the "Something went wrong with this webpage" error from occurring every now and then.
However, it does seem to have stopped another issue. Prior to this, in Avant 2012 and 2011 and even earlier -- possibly as long as a couple of years ago when I u[graded from 32-bit Windows XP to 64-bit Windows 7 -- I noticed when switching between tabs that some prior image of a previously-viewed web page would flash up for a fraction of a second. It has been less than a day since I switched of the GPU rendering option, but perhaps my video cards are not quite fast enough for GPU rendering instantaneous enough for the previous web page image to never be noticeable [sorry for the twisted sentence].
I got another clue about what might be causing the "Something went wrong" problem today, when I tried to open an external application and got the following error dialog:

- Too many user/GDI objects are being used by other applications
I have now stopped this VIA application from starting up automatically at Windows boot, and will load it occasionally only when needed, thereby freeing up quite a lot of User/GDI Objects table space.
I reckon that Avant 2010 Ultimate in Firefox rendering mode, with 20 or 30 tabs opened, might have been hitting the Windows overall limits for User and GDI objects, which is a MAJOR limitation of Windows architecture (and I'll bet it's a limitation that unfortunately Microsoft won't bother to remove in Windows

. My system has 8 GB of RAM installed, and rarely uses more than 50 to 60 percent of this, leaving always 3 or 5 GB of free RAM, but the USER/GDI Objects limit doesn't make any use of the spare RAM.
Now I can only wait for a few days to find out whether the intermittent "Something went wrong with this webpage" error has been done away with.
Running Windows task manager I found an application (VIA's audio support program VDeck) that was consuming a HUGE amount of User and GDI objects: