Hi,
It is very likely the results limit – 2000 is extremely high. WordPress needs to query all of their titles, contents, images and whatever fields needed. The search query itself is probably okay, it usually does not matter how many rows are queried on the database level. The issue in this case is more likely the post-processing – getting the result titles, fields etc.., which needs to be done via wordpress core.
For the live results I would not recommend going anywhere over 50. Most wordpress archive pages show 10 results at the time by default for the same reason – querying thousands of items at the same time can be slow.
Reducing the results limit below 50 should give you a very significant boost on the performance. Other than that, you can read up this knowledge base about performance improvements. However those target the main search query to be optimized as much as possible, but in your case I am almost certain, that the number of items queried at the same time is causing the bottleneck.