Well, it kind of depends on your definition of “watch TV” and “moving.”
Let’s assume for a minute that you’re talking about over-the-air antenna TV and the car is going at something like a normal speed. Then, unfortunately the answer is no. Digital signals don’t work in moving vehicles because they’re designed to get “locked in” in a way that just isn’t possible without standing still, or being close to standing still. This is a compromise that was built into the digital TV system when it was designed in the early 2000s. At that time it really wasn’t possible to send a digital data stream that would survive some of the things that happen to a signal when you try to receive it when it’s moving.
When you’re moving relative to a signal, there’s a slight drift to the frequency as you pick it up. If you’re old enough you might remember that there was a time when you had to turn the tuning knob as you got further away from the broadcast tower. Today that stuff is corrected digitally, but when you’re talking about the technology that existed when digital TV was invented, it wasn’t possible. So you can lock in no matter how far away you are (as long as there’s a signal) but you can’t lock in if you’re moving.
The threshhold is about 15mph. If you’re going slower than that you can probably watch over-the-air TV. So, if you’re thinking you might catch your favorite shows while riding a bike, you’re probably ok technically. (I still wouldn’t watch TV while bicycling, it seems like a dumb idea.)
The other side of the equation is what’s meant by “watching TV.” In 2015, it’s quite easy actually to get a digital data stream as you’re moving… using cellular data. It’s designed to deal with the problems caused by moving from place to place. When digital TV was designed, the fastest digital data was about 10kbps or .01Mbps. Today you can get 20Mbps in a moving car, 20 times faster. And that’s fast enough to stream live video.
We used to use the term “watch TV” to mean watching a live broadcast from an established broadcast source. More and more that term means streaming video from any of thousands of web sites, possibly millions, and most certainly you can do that in a moving car (please don’t do it if you’re the driver.)
Over the years there have been many attempts to create a system that would let you watch live, over-the-air TV from a moving car but every single one has either failed outright or simply not caught on because people don’t want to do that. They want to stream, and they’re already doing that.