Radial velocity surveys provide evidence that giant extrasolar planets are common, but their detection space is limited to only a few astronomical units from the stars. In order to close this gap, the adaptive optics assisted NIR imager NAOS-CONICA (NACO) at the VLT was used for a deep (15-20 minutes exposure time per target) L-band survey of a sample of closeby young stars. All stars are members of the Tucana and $\beta$ Pictoris moving groups apart from the the somewhat older star HIP 71395 that has a radial velocity trend suggesting a massive planet in large orbit. The chosen observation wavelength is very well suited for very high contrast imaging of close companions at this age and makes this survey unique. The goal was to detect substellar companions to these stars at distances as close as 5-20 AU and ultimately to detect giant extrasolar planets down to a few Jupiter masses, to measure their frequency, and - by comparison with models - determine their physical properties. This paper presents the results obtained on a subsample of 12 stars that have been observed during ESO P73.