Into represents transition or getting into something.
Into answers the question "where?" and is usually followed by a noun.
Examples are the best way to understand this.
He snuck into her room.
She ran into the mailbox.
Let's check into the facts.
We stayed up late into the night.
The snow lingered into April.
We got into an argument.
They entered into a pact.
He went into the army.
She went into business.
He turned into a frog.
Two into four is two.
I've been getting into baseball.
He's just not that into you.
In to is part of a verb phrase and answers the question "why?", usually followed by a verb.
She ran in to spike the ball.
She looked in to see what was going on.
He stayed in to avoid the rain.
My boss sat in to audit the meeting.
She came in to collect the papers. (But, she came into the room to collect the papers.)
I went in to buy some milk. (But, I went into the store to buy some milk.)
Sometimes, both into and in to will work. You could argue the entering into the computer.
He logged in to/into his computer.
With the verb phrase turn in, the meaning gives away the correct usage. The subject of the sentence did not actually turn into police, or a driveway, or a teacher.
They turned themselves in to the police.
The car turned in to the driveway.
She turned her homework in to the teacher.
The verb phrase give in is tricky. Into connotes compliance, willingness. In to connotes resistance. I think gave in to is the most common usage because usually it implies resistance, then giving in.
She gave in to temptation.
He shouldn't give in to his desires.
He gave into the pain. (as if masochistic)
*Usually the answer will be into. But if you have turned in or gave in, it's worth looking at it closely just to be sure.
*Always check the entire document for consistent usage.
This is just a quick summary with examples. See these sources for more details and logic.