There are a few different accomodation options:
1. The university provides accomodation for incoming students at the students dorms. For more information visit the University Website
2. olx.ro is a website for various sales ads, including rent options
3. There are a few facebook groups where you can post an accommodation ad or reply to one.
For quick reference, most students usually look for places in cornisa(where the university is), 7 noiembrie which is close by, or centre, which is a maximum of 10-20min away. See the Targu Mures city map here!