Wobbly on my new roadbike

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Also ... relax, d@mn it! Sounds counter intuitive but the death grip on the bars makes you less stable. Fluttering your fingers or playing piano on the bars may keep your hands relaxed.

Yup. And making sure that your elbows are not locked. DH drilled this concept into my brain early-on. Think of your arms as shock-absorbers. Your upper body will thank you.