I had an Aspire One D270 laptop with a 32-bit Intel Atom CPU and 1 gigabyte of RAM, so I installed Debian with Xfce on it, but even then it’s running way too slow.
Is there anything I can do to make the laptop faster and more responsive given its limited memory?
You need something like DamnSmallLinux, not Debian. Debian users about 800 MB of RAM with XFce, on a clean boot. It requires a minimum of 2 GB with a modern browser (one tab, 4+ GB with more tabs). DamnSmallLinux uses about 128 MB RAM on a clean boot, and with the Netfront browser about half a gig. Definitely better for such a laptop than any modern distro.
Antix linux would also work great, and DSL is based on it.