I used Bridge for years before Adobe came out with Lightroom. I started using LR almost immediately after its introduction, but it did not usurp Bridge from my workflow.
I am a huge fan of LR because of its ability to perform reasonably powerful image searches that can span multiple hard drives. I also use LR every day because of its ability to handle huge numbers of ACR presets, powerful capabilities for producing prints, photo books, slideshows, etc. Sure, there are more powerful, separate programs to produce prints, or books, or slideshows, and I own some of these programs, but for me LR does 95% of what I need in these areas in one nice, seamless package.
Bridge is a very different programs. Think of it as a conventional file manager that was put on steroids to make it perfect for *really* fast browsing of images within a directory, comparing images, quickly assigning and obtaining info about each image, dispatching them to other programs to be worked on, etc. Bridge's major limitation is that it has no serious / fast search facilities that span many directories or several hard drives, because it has no pre-constructed database.
In contrast, the file handling aspects of LR are much more along the lines of a DAM, a Digital Asset Management program. It provides reasonably powerful search facilities, but the trade-off for this is that you can't just take a quick look at a bunch of files: they first have to be ingested, ie, metadata extracted and placed in LR's database, preview images generated, etc.. Sometimes it's just not worth it to spend the time/energy go through this and it's easier to use Bridge for this. LR is also annoying when you want to move files around -- you absolutely must do this within LR. If you forget and move the files outside of LR, LR will lose track of where they are, and then, you have to go through an annoying procedure to tell LR their new location, re-synch the directory to make sure you didn't add or delete anything, etc. Also, I find LR's display of the directory structure horrible compared to Bridge.
I use both programs virtually every day. My workflow is this: Using Bridge, I transfer images from my cameras to my computer. I'll also make a backup copy of the files, do initial culling and sorting into directories, assign keywords, stars and color ratings, etc. all within Bridge. Then, after the locations of the files are relatively stable, I'll ingest the result of Bridge's preliminary file handling work (primarily, the "keepers") into LR for ACR-level processing.
Sure, in principle, I could do all the preliminary file handing in LR, but I find it much faster and less painful to do it the way I described.
HTH,
Tom M