DD had a pillow around 20 months as well for a while when she was in her crib. Switched to toddler bed at 25 months, and now is in a full bed at 3 years. She now has about 4 pillows on her bed and likes to sleep with all of them.
We gave him a pillow in his crib at around age 2. He never slept on it (it was a toddler pillow) but snuggled with it some. We converted from the crib straight to a full sized bed the month before he turned three and gave him a regular sized pillow. He often starts out sleeping on it and then eventually turns sideways in the bed.
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Re: converting to toddler bed
We gave DD1 a pillow (an older one that was pretty flat) when she was around 20 months I think & still in her crib. I say go ahead & give her one!