We are massively subsidizing livestock production in Europe!

There is actually no direct link between livestock production and EU subsidies. The direct payments of the CAP are linked to the land, meaning that you have to justify a certain amount of farmland to receive subsidies. If you take production of pork or poultry for example, these livestock systems do not necessarily require farmland and therefore those sectors are not directly subsidised. 

Milk or cattle production is different, as pasture land or land for fodder are necessary and these receive direct subsidies. Those pasture lands are very often placed in zones where it is not possible to grow crops so this contributes to the maintenance of the landscape. 

Some interest groups believe that indirectly, the livestock sector is massively subsidised through the amounts received by crop producers, knowing these crops will be used for feeding the animals[ref]https://www.greenpeace.org/eu-unit/issues/nature-food/1803/feeding-problem-dangerous-intensification-animal-farming/[/ref]. This is forgetting that an important part of the ration (especially for non-ruminant) is soybean which is only produced in the EU in relatively small quantity and is therefore imported from North or South America. In addition, an important part of these crops fed to animals are by-products (oilseeds) as the main element of the crop is used for human consumption (e.g. when you grow sunflowers, you use the oil for human consumption and the by-products are used to feed animals). Therefore, using the argument that livestock production is largely subsidised for the production of feed is denying that most of these crops are firstly used for human consumption and that animals are just eating the by-products that would otherwise be wasted if not consumed by those animals.