Today's HTML is not your Grandpa's HTML any more :-D While HTML hasn't changed ALL that much (you can look them up at w3c.org,) there are also now cascading style sheets (CSS) which are not supported, AFIAK, by any 68k Amiga browser.
When I used to browse the Internet on my modem-based Amiga, I had problems mostly due to the serial device I was using versus the modem. Others at the time reported differences in the TCP/IP stack. Now, these weren't directly related to Amiga.org, but general Internet usage overall.
If you have a box Amiga out there in the work shop, see about getting a serial card rather than using the Amiga's internal serial. You'll have a faster and more reliable serial device that way.
If you're using an A1200 out there, get a Netgear 802.11b network card and set up a wireless access point or router in your house. I have used this setup with Miami and AWeb before with no problems.
If your workshop is close enough to the house to run a phone line, put a network card in your work shop 4000 (steal one from the house or get one online) and replace your phone cable run with some CAT5 cable, which will easily carry your phone line and your network. (The maximum length of a CAT5 segment is 100m, end-to-end.)
As for the problem with animated GIFs on the site, use a datatype that allows you to turn off animations or doesn't support animation at all, or disable GIFs in your browser. Most useful images are JPG or PNG, anyway. When I was still using my WarpEngine 4040 with my PIV, animations on websites worked fine, even here.
And finally, have a look at your phone line. There may be some interference on the line causing problems with your page loads. Being able to talk on the phone without hearing noise is irrelevant as we cannot discern phase shifts that the modem can. Try testing the dial-up at other points in the house, assuming this phone line is connected to the house wiring. Disconnect extraneous devices and try the dial-up.
Just for shyts and giggles I put my 060-based 4000 online with a USRobotics 56k Sportster. Forum works fine with the latest demo release of IBrowse, Warp Datatypes, and Miami. Being that I do not know where you are (and hence the quality of your phone system -- some countries still use twine rather than 26 gauge copper or better) or at what speed you're connecting, I do not have much else to offer.
So onto the next curiosity. If someone running a simple dial-up BBS would be helpful, maybe you could set on up. Run it on a machine with network and dial-up capability and offer two or more nodes (via modem and telnet.) Attract some users there and there you go. Offer a front-end to news groups and help get them livened back up. Offer text-based browsing (ala Lynx) which works very well (latest version on Solaris) with A.org, with the exception of bad cookie roots, and you will miss out on the emoticons most of the users use in posts, so you may take some posts out of context.
For that matter,
http://www.molgen.mpg.de/~alynx/Okay, that's all I have. Hope things work out for you. This forum is exceptionally valuable.
