As our bibs have to be very absorbant, I cut 2 bibs, 1 out of fleece(usually use this side for lining, but not always) and 1 out of cotton or poly/cotton. That way I don't have to bind the edge and they can be reversible. I leave an opening for turning, after sewing right sides together and topstitch edge with a decorative trim, catching the opening with the topstitch to close. It does take twice the fabric but it goes so much faster and you don't have to cut your bias tape or seam binding.Maybe if you know how to look up, in projects, you can see the ones I have made for Avery, my granddaughter. Good luck-Toogie

The link below didn't work so here it is. I always sew wrong sides then turn my bib's inside out. Are you using a binding foot? I am getting one for xmas I hope LOL My daughter in law kinda mentioned a new foot LOL I have the adjustable one and it does good on straight items I'm not to good with the curves on it though.
add link for one I did

I have many TONS of baby bibs and the best way I came up with was to sew the outer edges together (right sides together). Turn right side out, so now you have the raw edge being the one that goes against the neck. That's the one I would sew extra wide double fold seam binding on. I would then tie a know at both ends.

Seam binding does not have much 'give' so it is not easy to shape round curves - though it is find on straight runs. Bias binding (which you can make yourself) is usually made of cotton or cotton/polyester and can easily be pressed round curves. Look at ready-made bibs for ideas..

I use bias binding or line them.

Thanks for all your answers - will have another go.