Data split -- Creating a copy of database without outage

Can you please help with advice?

I need to design a solution for a database which will grow and will require
a horizontal split at some moment.

Here is how I am planning to do it: Every record in every table has a shard
number. After the database becomes too large (gets too many requests), the
tables need to be horizontally split. It means that for every table all
records with some specific shard numbers need to be moved to a new database.

My plan is to (1) create a copy of the database on a new server
(backup/restore?), (2) synchronize the databases (how?), and then (3)
forward all requests for the moved shards to the new database. (I will get
rid of the unneeded shards later). An important requirement: there should be
no outage for the users.

What are the ways to achieve it?

Thank you in advance,
-igorS