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The SINFONI survey for Unveiling the Physics and Effect of Radiative feedback (SUPER) aims at tracing and characterizing ionized gas outflows and their impact on star formation in a statistical sample of X-ray selected Active Galactic Nuclei (AGN) at z$\sim$2. We present the first SINFONI results for a sample of 21 Type-1 AGN spanning a wide range in bolometric luminosity (log $\mathrm{L_{bol}}$ = 45.4-47.9 erg/s). The main aims of this paper are determining the extension of the ionized gas, characterizing the occurrence of AGN-driven outflows, and linking the properties of such outflows with those of the AGN. We use Adaptive Optics-assisted SINFONI observations to trace ionized gas in the extended narrow line region using the [OIII]5007 line. We classify a target as hosting an outflow if its non-parametric velocity of the [OIII] line, $\mathrm{w_{80}}$, is larger than 600 km/s. We study the presence of extended emission using dedicated point-spread function (PSF) observations, after modelling the PSF from the Balmer lines originating from the Broad Line Region. We detect outflows in all the Type-1 AGN sample based on the $\mathrm{w_{80}}$ value from the integrated spectrum, which is in the range 650-2700 km/s. There is a clear positive correlation between $\mathrm{w_{80}}$ and the AGN bolometric luminosity (99% correlation probability), but a weaker correlation with the black hole mass (80% correlation probability). A comparison of the PSF and the [OIII] radial profile shows that the [OIII] emission is spatially resolved for $\sim$35% of the Type-1 sample and the outflows show an extension up to $\sim$6 kpc. The relation between maximum velocity and the bolometric luminosity is consistent with model predictions for shocks from an AGN driven outflow. The escape fraction of the outflowing gas increase with the AGN luminosity, although for most galaxies, this fraction is less than 10%., Accepted for publication in A&A, 34 pages, 14 figures and 5 tables