turn up the heat on - Phrases/Idioms
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Also, put the heat or screws or squeeze on; tighten the screws on. Pressure someone, as in The cops turned up the heat on drivers who show signs of drunkenness, or They said they'd tighten the screws on her if she didn't confess. All of these slangy terms allude to forms of physical coercion or torture. The first dates from about 1930, the variants using screws from the mid-1800s, and squeeze from the late 1700s.