S.B.
Well, you need to stop telling her you're putting the dog to sleep. You're not. You are euthanizing the dog. A child that age doesn't understand the phrase "putting to sleep" and may think that it's possible that she herself may never wake up if she goes to sleep.
We told our daughter last year when she was 5 that our dog got sick, and the doctor couldn't make her better, so she died. We told her that when people and pets die, they go to live in heaven with God and wait for us.
Yesterday her Grandpa died, and she said that he was waiting in heaven with Tootsie now. We're all sad, but that's a nice thought.
If you don't believe in heaven, or that pets go to heaven, I'm not sure what you should say, but that is how we handled it. We also made sure to let her know that it's ok to be sad and it's ok to cry. She may not fully understnad and want to just go back to playing, she may ask lots and lots of questions. I'm sorry that you have to go through this, I know it's hard. Our daughter will still, a year later, randomly tell us that she misses her dog and wishes she didn't die. We just let her know that we wish that, too.