People are making the assumption that high density=high-rise tower blocks, which is not true at all. Take a look at Paris and its 4-5 storey block heights, which still manages to eek out the highest population densities in Western Europe, and yet is considered to be very livable.
I've wondered if there was a way of creating a bungalow with a town house (+3 storeys) on top.
It would then be possible to have a green roof garden on top of the "bungalow" for the town house whilst still having a garden at ground level for the "bungalow".
Other options could be 4 storey duplexes (think traditional house layout, just with one stacked on another).
With many of these options parking becomes the issue. However, it would be possible to create basement parking (including under open space, which could be plazas and/or playgrounds. Add in some nearby allotments and there could be some decent outside spaces.
Although often the challenge is to get people to accept that they don't need a garden as large as they think that they do. For example a lot of our desire for outside space is to have space for the children to run around, if there was a playground within sight of our home it could make it less of a need for us to have so much of our own space.
However with higher densities it could reduce the need to travel by car. For example, where I live the most anyone needs to walk to a school or supermarket is about 1.5km, if they was reduced to 800m then the need for car ownership reduces significantly. Not only that, but buses become more viable as each stop is closer to more people.
It also means that big item facilitates becomes easier to justify.
Previously I've compared the population of Minehead with the capacity of the Butlins and it's shocking just how many people you can fit in to quite a small area. That's with a maximum of 3 storey building for the vast majority of the Butlins site (there's one 4 storey building of flats) and no flats above shops, etc.