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Yes, this is possible. Consider the intersection of the helicoid $z = \tan^{-1}(y/x)$ with its tangent plane $z = y$ at $(1,0,0)$. The projection of the intersection curves to $\{z = 0\}$ consists of the line $y = 0$ and the curve $x = y/\tan(y) \sim 1 - y^2/3$. After a rigid motion so that the tangent plane is horizontal and the tangent point is the origin, the union of $\gamma_1$ and $\gamma_2$ will be a $C^{1/2}$ graph $\{(t,\,\varphi(t),\,0)\}$ where $\varphi(t) = 0$ for $t \geq 0$ and $\varphi(t) \sim (-6t)^{1/2}$ for $t < 0$.