Can I root my Nexus 5 without a custom image?
29 January 2015: TWRP can no longer install official updates past 5.0 on Nexus devices. Updating my language here to say "install unofficial updates" instead of "official".



Since you specifically mention TWRP in your question, I'll just use it as my example. Note that CWM or any other possible Android Recovery project will also apply.

It looks like there's a slight misconception at play. I'll just drop a few clarifying notes here:


TWRP is indeed an image to be flashed.
Flashing TWRP does not overwrite your bootloader.
TWRP is not a bootloader replacement - it is Recovery software. It is a drop-in replacement for Android's stock open-source Recovery Mode, and therefore has the access needed to do things like


Create/Restore backups
Root devices
Install unofficial (and some official) updates



So the short answer is: Yes! It is indeed possible to root without overwriting the stock bootloader!

A few more notes:


TWRP (And CWM Recovery) is a mature (Several years old and quite stable) Open-source project that is used by a huge portion of the Android community and considered very trustworthy.
Flashing TWRP will replace the contents of /recovery/.
You generally need to use the stock bootloader in order to select the boot-to-recovery option in order to use TWRP.