Learn English – Should we use a comma, a semicolon or a dash before “trust me”

If we set aside the structure of the sentence (which I think could be improved), you can put an em-dash there, but if you do, you should also replace the comma at the end of the clause, thusly:

In conclusion, the reading claims that minimum wage is harmful to the companies, which the lecturer contradicts by providing three points—which form a compelling argument—to illustrate his viewpoint.

Putting em-dashes there makes it sound as though you are surprised that the three points form a compelling argument.

If you use commas instead, they should also match:

In conclusion, the reading claims that minimum wage is harmful to the companies, which the lecturer contradicts by providing three points, which form a compelling argument, to illustrate his viewpoint.

In this instance, commas are appropriate because the clause is parenthetical. However, I don't think you really mean to have a parenthetical clause there, so you might try something like:

In conclusion, the reading claims that minimum wage is harmful to the companies, which the lecturer contradicts by providing three points that form a compelling argument to illustrate his viewpoint.

Which would avoid the issue entirely.

I don't really think any version is appropriate in "formal writing" (unless it's reported speech).

The comma indicates a stronger link between the two statements, so we naturally interpret it as meaning "You can trust me when I say that I don't give a damn".

Using a semicolon implies that the two statements are logically distinct, so we're more likely to assume the speaker means "I don't give a damn [about whatever was just mentioned]. You can just trust whatever I say in this matter".

Both the above are just tendencies. It's still possible to adopt the "contrary" interpretation. But to my mind, the dash gives no real support to either alternative. So it's probably the worst option.