What age did you start tummy time?

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I have a playmat for my baby and he is fine on his tummy and can lift his head briefly and turn it. I'm terrified of tummy time because of suffocation. I know it's good for them but I don't want to start it until he is much older (currently a month old) and I think when he is 3 months or 5 months I might be okay with it?

I had someone tell me he needs to start it now since his head movement is advanced and he needs to keep practicing.