Do you have any advice mother to mother for consequences for my 5 year old?

I have a 5 year old daughter and a newborn. Our 5 year old starts school this year and has never really been socialized as she never has been in daycare or been around kids - except a few visits with her cousins and she had fun playing with them. I say this because sometimes she still has trouble communicating things, especially feelings in the moment. She was finally going to start a daycare like program last year but COVID - so she’s still home with me (I’m a stay at home mom). She’s very polite with our family and friends and 90% of the time is well behaved and fun.

I’ve noticed since back when I was pregnant and talking about a new baby a few months ago her tantrums picked up in intensity. She started hitting, screaming, having an attitude in general more. For discipline she has 5 minute timeouts and we’ll take toys for 24-48 hours if she keeps up the behavior. After awhile once she’s calm we’ll sit and talk with her when she’s level headed and she apologizes for throwing a fit or being disrespectful. Since we had the baby she has been super helpful and loves handing us stuff when we’re changing diapers, holding her sister, and when she cries she always asks what the baby needs and if she’s okay.

Today we were laying in bed with our newborn, she was sleeping. I dont remember why she said it, I think I spilled a little water on the bed. My 5 year old then said she wanted to spill cold water on my husband and our sleeping newborn he was holding because it would “make her happy”. We shut that down and explained that would be mean and hurt the baby. She went to timeout for a bit and then once she came back and was calm, we talked to her and she apologized and said she wouldn’t ever do that as she wouldn’t want to hurt the baby.

How can I teach her more about empathy and WHY it was bad to say that? I worry that me sending her to timeout was the wrong move, should I have sat in the moment and explained it? How can I explain to her that it’s not just “mean and bad” but that we all love her baby sister and need to be gentle and not say things like that in general? Should I have distracted her and not put so much attention on what she said? Should I have done more besides timeout? Any advice? I have no female motherly roles to help me out.

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