I have a database that will have 1m+ records of the following shape:

create table simplified(
    url text primary key,   -- google.com/search?=something
    url_host text not null, -- google.com
    url_path text not null  -- /search?=something 
)

I understand that to search for a urls substr a simple query could be:

select * from simplified where url like '%google%'

I am trying to optimize for speed of search and am willing to form the data in anyway. Some questions I have:

1. Is the LIKE operator really what I want for speed? or is there a faster alternative?
2. Can tsvector or tsquery be implemented with a GIN index in a meaningful way for single string search?
3. If not #2 is there something similar?

Thank you for your time in advance.

I have tried tsvector implementations to no avail, and some research on the topic.