Overview
When creating an API like </api/_search>, I looked into how to create URL segments starting with underscores, so this is a personal note for future reference.
Method
The information was found in the following documentation.

The key point is:
To create URL segments that start with an underscore, prefix the folder name with %5F (the URL-encoded form of an underscore). Example: %5FfolderName.
I was able to resolve this by creating the file as /api/%5Fsearch/route.ts.
Summary
I hope this serves as a useful reference.