She lied to Rick to save Lizzie, then lied to Tyreese for the aforementioned reasons. It doesn't have to make sense, because this show is stupid.
Okay, if she lied to save Lizzie
why would she put up with Lizzie's bullshit for so long when warning-bells were there as early as the prison, then?
Let's say this is true: What is Lizzie's motivation for killing the prison's sick? Lizzie admitted she killed her sister for the sake of "showing" Carol they (walkers) aren't as dangerous as Carol and everyone else has told her they are. She was even close to killing Judith to prove this point.
Lizzie wouldn't/couldn't have known that the people that were sick were going to turn or not: What benefit would Lizzie have in doing that?
Carol's reasoning (to protect the group) makes the most sense; facing an unknown threat (the virus) with no means to stop it's spreading (outside a quarantine that may be too late to enforce as shown as the "series" progressed with that storyline) outside of possibly killing the carriers (which, IMO is stupid from a medical and logical perspective on illnesses, but roll with it) that would stop the infection possibly... why would Carol lie about that?
So basically: What is Lizzie's motivation in killing the sick beyond "she's crazy!!!!!!!!!!!!111!!111!1" that you're going with?
Further more: How would Lizzie have dragged the bodies of the two outside, given she's a
child and then set them on fire if she wanted to see them turn?
I don't disagree that there is some weird gaping logic plotholes that the writers do (hell the same happens in the comics at times), but this is one line of logic that simply doesn't connect with what we are shown/seen. Carol lying simply doesn't make sense. Her wanting to die by Tyreese's hand by admitting the murders? Sure, that's logical and seen by her giving him the gun. Her
lying to protect the girls? That's a
little too much of a stretch.