Does the current discourse support "partial match search" in topic's title?

It doesn’t work on my discourse.

For example, my discourse has the title, “approximately.”
When I search with “approx”, the suggested search results will show up.
When I search with “mately”, the suggested search results will not show up.

My discourse is old, v1.6.0.beta10 +9.

It does not matter, you need to upgrade ASAP, many exploits were patched since 1.6…

This is by design, common way to have search the way you are describing is to use trigrams. At the moment there are no plans for a trigram based search.