shamsudeen the Scout extension lets you use alternative search drivers. Those take the form of cloud services or additional self-hosted services with various system requirements https://discuss.flarum.org/d/30874-scout-search
Improving the MySQL-based search query of Flarum is complicated, any improvement to relevance would be to the detriment of performance. If you cannot host a separate search service, a MySQL-based separate search Index like the one provided by TNTSearch driver via Scout is probably the best solution, though TNTSearch itself doesn't seem very good. I know other forum software maintain their own keywords database to speed up search, it's probably something that could be implemented via an extension, or if it already exists in the Laravel Scout ecosystem, dropped via the Scout extension.
To be clear, Flarum's search bar is already natively instantaneous ("AJAX"). You are probably only referencing the fact that the default search driver is only capable of full word matching? Or are you expecting to see more when a result is found?