How can I remodel a bedroom into a family room?

First idea that comes to mind is installing a DYI or other shelving unit on the short wall at the far end of the room, if windows allow- leave the bottom shelves 'open' to insert bins or baskets for stowaway storage and/or an ottoman or bean bags chair. Then decide if there is a better wall for the TV, if you plan to install one, or art or a mirror if not. On the opposite wall, a smaller scale sofa/loveseat or a futon is a practical fix for seating. Brighten with throws & pillows to add a personal, cozy touch. Use a trunk for a coffee table if there's room (more storage) & a floor lamp with 3-way light bulbs. A fun carpet will add color & personality. You can make your own out of carpet squares from big box stores. Use a curtain rod with rings & little clamps for kids' artwork. You can change it out as desired.

Use your furniture to shape the room, don't feel pressure to just line the wall with them

You can do anything with any room. As it it narrow I would go with a corner couch to provide maximum seating without taking up heaps of floor space. I would then position the TV on the wall - again freeing up the floor beneath. Add a large floor rug that takes up most of the space between the couch and the opposite wall and a coffee table that doesn't dominate e.g. a clear glass table. Add a few plants and perhaps a mirror to reflect the light, maybe one medium sized arm chair on a diagonal to the couch - but still part of the seating hub. Add some colour if you need to in cushions, perhaps a feature wall and curtains, but keep the rest of the room white and light to make it appear larger. Best of luck.

I love Bernadette's ideas. Depending on the size of the room and where the door is located, you can remove the door and widen the doorway to make it feel more airy and open.