No, it doesn’t. A 32 bit OS has a limit of 4 gigabytes of memory unless it’s using a PAE (physical address extension) kernel. Windows XP actually DOES have PAE available, but they don’t license the OS to work with it, so they tell you that you’re stuck unless you go with 64 bit.
I use 32 bit Linux Mint and I have 8 gigs of ram, fully usable. Many use the 64 bit version instead of installing PAE, but I’ve never noticed a real speed difference and I found the compatibility issues of 64bit to be annoying to deal with a few years ago and never went back.
“...unless it’s using a PAE (physical address extension) kernel....”
Right you are. PAE adds 4 address bits for a total of 36 bits. That’s 64GB on a 32 bit PAE linux kernel, 4GB on a standard 32 bit kernel.
I’ve been using a 64bit kernel since around 2002. It was very painful at first but support has vastly improved in recent years.