Web browsers running heavy applications ("web sites" we used to call them) will simply be slow because of them. Ad-blockers are a must since plenty of ads are javascript-heavy, and you can surely switch to NoScript-approach to disable all JS. Of course, not more than a few tabs running in the background.
Pre-emptive multitasking should allow your system to run multiple programs just fine, provided you've got a sufficiently fast core in the first place.
My Sony Vaio Z790 from 2009 with Intel Core Duo P9700 runs just fine (though that has an SSD as well): it's dual core score is identical to it's single core score on cpubenchmark.net at around ~1100, while the fastest Atom never reached more than 1000 single-core score until Q2 2021 (with x6211E), and there are still only 5 of Atom CPUs that beat that, and another 4 topping 800 ever. So your netbook likely has a super slow single core with low cache too.
At the time of the netbook craze, I've invested in one of the fancy metal HP "business" netbooks on the expensive side (~$500): lovely 10-11" machine, but slow as hell. I quickly sold it away after basically not using it at all.
Edit: I just looked it up, and it was HP Mini 2140 with a feisty N270 Atom CPU scoring an entirety of ~170 on cpubenchmark in both single- and multi-core benchmarks.