Just tested it and didn't find it super bad, but I am honestly (negatively) surprised. It does indeed lag quite a bit and feels very unnatural. Chrome here as well. I use React quite a bit myself and haven't experienced this in my own apps. Odd!
There's also a slightly annoying flicker when you type the last bit and it keeps returning the same movies.
Something like that certainly plays it's role. I'm doing that as well for certain inputs and experience no lag at all, so it depends more on their implementation of the store. I think their search store is probably doing a lot more than mine (although, entering text, querying an API, processing response async). It seems like they could introduce some optimizations there.
There's also a slightly annoying flicker when you type the last bit and it keeps returning the same movies.