I am wondering what is more efficient and faster in performance:
Having an index on one big table or multiple smaller tables without indexes?

Since this is a pretty abstract problem let me make it more practical:
I have one table with statistics about users (20,000 users and about 30 million rows overall). The table has about 10 columns including the user_id, actions, timestamps, etc.
Most common applications are: Inserting data by user_id and retrieving data by user_id (SELECT statements never include multiple user_id's).

Now so far I have an INDEX on the user_id and the query looks something like this

SELECT * FROM statistics WHERE user_id = 1

Now, with more and more rows the table gets slower and slower. INSERT statements slow down because the INDEX gets bigger and bigger; SELECT statements slow down, well, because there are more rows to search through.

Now I was wondering why not have one statistics table for each user and change the query syntax to something like this instead:

SELECT * FROM statistics_1

where 1 represents the user_id obviously.
This way, no INDEX is needed and there is far less data in each table, so INSERT and SELECT statements should be much faster.

Now my questions again:
Are there any real world disadvantages to handle so many tables (in my case 20,000) instead of using of using one table with an INDEX?
Would my approach actually speed things up or might the lookup for the table eventually slow down things more than everything?