Hi Jules-Aymeric,
Welcome to the Apple Community,
where some of us do our best to help each other troubleshoot :-)
Wondering if your MacBook Air has enough memory / storage free (at least 50Gb),
for Safari to operate smoothly.
Apple support have an article online that may be helpful:
How to check / free up storage space on your Mac:
https://support.apple.com/en-us/HT206996
Sometimes third-party (non-Apple) software can cause issues.
(Eg: Try disabling any ad-blockers, or unnecessary Virus checkers,
or
update third-party software to be compatible with Monterey.
or
If you wish to delete old third-party apps, best to contact the developer for the uninstaller)
Also, you may want to browse this article:
If an app freezes or quits unexpectedly on Mac
https://support.apple.com/en-au/guide/mac-help/mchlp2579/mac
Hope this helps :-)