Before I could grab hold of him, he'd ________ for the exit.

Make a break (from/for) (phrase): 
English Meaning - To escape from/towards somewhere or something, often by force. 
Bangla Meaning - পলায়ন করা।

Example Sentences: 
- A group of prisoners made a break from the jail a few years back.
- The cat made a break for the door.

Complete Sentence - Before I could grab hold of him, he'd made a break for the exit. 

Other options, 
Make a bomb - (প্রচুর অর্থ উপার্জন করা) To make a lot of money. 
Make a beeline for something - (দ্রুততার সহিত একদিকে সোজাসুজি গমন করা) To move quickly and directly toward something. 
Make a bed - (বিছানা করা/ পাতা) To pull up and tidy the covers on a bed after it has been slept in. 

Source: Cambridge Dictionary.