You can install & use a third party browser. However, even many of the third party browsers are now only (or will only be) supporting the latest three versions of macOS just the same as Apple. The main reason this occurs is how Apple tends to change the core framework of macOS and their XCode software.....Apple only has focus on the current & future, not on older hardware or operating systems. It is much more work for developers to support older systems under these conditions. Some third party browsers do have long term support (LTS) versions which may still be supported on the older versions of macOS for a while. Sometimes someone may even backport changes to older systems, but that is hit or miss since it is a lot of hard work.
I'm not sure which third party browsers are still available for Catalina. Firefox may be one since I know they tend to have LTS versions. Forget Chrome.
I did find this notice from the Vivaldi browser developers about the last supported version for macOS 10.15 as well as their explanation.
Here is the download link for Vivaldi v6.9 (from Aug. 2024 so it will probably work well for a while until the Internet changes some more):
https://downloads.vivaldi.com/stable/Vivaldi.6.9.3447.35.universal.dmg
I did see a link for a snapshot version v6.10.x which still seemed to have a link for Catalina, but I could not find a direct download link for the stable channel version (aka the official release version). I'm not sure what version will be applied if you try to use the update function within the browser....will it update to the last supported version compatible with 10.15? or will it update/offer an incompatible version? I don't know, but the Vivaldi developers did say updates compatible with Catalina would occur through October 2024.
FYI, Vivaldi is based on Chromium which is also used by Google Chrome, but Vivaldi offers more customizations & extra features.
Firefox still seems to be fully supported on Catalina with the current release:
https://www.mozilla.org/en-US/firefox/132.0.2/system-requirements/
Firefox Download:
https://www.mozilla.org/en-US/firefox/new/
Edit: I removed Vivaldi link to article regarding Catalina because it's contents violate the forum Terms of Use.