How can I create an index on a substring of a text column?

You could create a computed column that persistently stores the information:

alter table table_name 
    add column text_has_slash tinyint
    generated always as (text_col like '%/%')
    stored
;

Or, if you want to treat null values as negatives:

    generated always as (coalesce(text_col like '%/%', 0))

The column value is computed and stored in the table (it is automatically updated by the database when the value changes).

Now you can use that column in your query:

select * from table_name where not text_has_slash;

Demo on DB Fiddle

Filtering on a pre-computed value should already improve performance.

Creating an index on a boolean column does not necessarily help, because there are only three possible values (0, 1, null). Unless the values are very unevenly distributed, it is often faster for the database to perform a full scan. On the other hand, you might want to include this column in a multi-column index, if you have more search criteria than those you have shown.

The real issue is whether the entire table needs to be checked versus whether there is some way to limit the number of rows via an index.

First, let's decide whether such an index will even get used. As a Rule of Thumb, if more than 20% of the table is matched by some index, that index won't be used. (The "20" varies depending on the phase of the moon.) The logic is that bouncing between the index's BTree and the data's BTree costs something. Such bouncing is worth the effort if there aren't many rows -- that is when the index is "selective".

So, if more than 20% of the rows have "/", none of the suggestions will be efficient. LOCATE may be more efficient than LIKE; a REGEXP is probably slower than either. Still, the main cost in the query will be having to look at every row.

If, on the other hand, very few rows have "/", then any of the pre-computed indexes will be beneficial.

If the real test is WHERE x LIKE '%/%' AND ..., then we need to look at the secondary part of the test. It may be that even a non-selective test for "/" can be effectively combined with the other part of the test.

Bottom line: Give us the complete picture, plus some statistics.

Maybe LOCATE can help you.

WHERE LOCATE('/', table_name.text_col) = 0 

When LOCATE returns 0 it means that substring isn't find in string. More info at https://dev.mysql.com/doc/refman/8.0/en/string-functions.html#function_locate.

You asked for LEFT(), which isn't what you're looking for. This function returns substring from beginning of string. Syntax is simple, https://dev.mysql.com/doc/refman/8.0/en/string-functions.html#function_left