Why doesn't Serial work as I'm expecting?

Projects Programming
1

I'm trying to figure out how Serial works.
I wired together pins 0 and 1 on my UNO. So, in theory, I expect everything I send via Serial.write to come back via Serial.read.
Why, then, if I execute the following code, is the TX led on and the RX led off (I'd expect both to be on)?

void setup() {
  Serial.begin(9600);
}

uint8_t count = 0;

void loop() {
  Serial.write(count++);
  Serial.read();
}
2

The LEDs on the Uno are driven from the USB-to-serial converter chip, so the Rx led will only flash if data is received via the USB.