The Taylor-Hood approximation of the Stokes flow is a mixed finite element method, for which error estimates generally have the form
$$
\|u-u_h\|_V + \|p-p_h\|_M \leq C (\inf_{w_h\in V_h}\|u-w_h\|_V + \inf_{q_h\in M_h}\|p-q_h\|_M),
\tag{1}
$$
where $(u,p)\in V\times M$ is the exact solution and $(u_h,p_h)\in V_h\times M_h$ is the approximation. For the Stokes flow, $V=H^1_0(\Omega)^d$ and $M = L^2(\Omega)$ (with mean value zero). For the $P_2$-$P_1$-Taylor-Hood element, $V_h$ consists of continuous piecewise quadratic polynomials and $M_h$ of continuous piecewise linear polynomials, for which both terms on the right hand side can be bounded by quadratic approximation errors using standard arguments (e.g., Bramble-Hilbert lemma and transformation rules):
$$
\begin{aligned}
\inf_{w_h\in V_h}\|u-w_h\|_{H^1} &\leq C h^2\|u\|_{H^3}\\
\inf_{q_h\in M_h}\|p-q_h\|_{L^2} &\leq C h^2\|p\|_{H^2}\\
\end{aligned}
$$
(rule-of-thumb "number of derivatives on the left $+$ powers of $h$ $=$ number of derivatives on the right").
Inserting this into $(1)$ yields
$$\|u-u_h\|_{H^1} + \|p-p_h\|_{L^2} \leq C h^2(\|u\|_{H^3} + \|p\|_{H^2}).$$
(assuming that the exact solution is in fact regular enough).
Since both the continuous and the discrete Stokes problem are an upper triangular coupled linear system of the form
$$\begin{aligned} Au + B^* p &= f\\ Bu &=0\end{aligned}$$
the error estimate (which exploits that the difference of the solutions satisfies a similar system for $A_h$ and $B_h$ and uses the invertibility of $A$ on the kernel of $B$) for $u-u_h$ depends in general on the error in $p-p_h$.
However, if you look at the proof, there's a loop hole: If the null space of $B_h$ is contained in the null space of $B$, then the coupling term in fact drops out, and you obtain an error estimate involving $u$ only:
$$\|u-u_h\|_{H^1} \leq C h^2 \|u\|_{H^3}$$
From there, you can apply the standard Aubin-Nitsche trick (if the adjoint equation is well-posed, which is the case if the domain $\Omega$ is regular enough -- a convex polygon in 2D or has a boundary which can be parametrized by a Lipschitz differentiable function) to obtain a convergence rate for the $L^2$-error of one order higher:
$$\|u-u_h\|_{L^2} \leq C h^3 \|u\|_{H^3}$$
You can find these results in Ern, Guermond: Theory and Practice of Finite Elements, Springer, 2004. (The error estimates are collected in Theorem 4.26, while the necessary regularity for $\Omega$ is defined in Lemma 4.17; the proofs are unfortunately scattered across the book, and I think $\ker B_h\subset \ker B$ is nowhere verified explicitly.)
