Somerset Council defines a conservation area as an area "of special architectural or historic interest, the character or appearance of which it is desirable to preserve or enhance", and its own page lists "demolishing any buildings or structures (including outbuilding, gates, fences, railings, walls)" and "laying paving or building walls" among the works for which its written consent may be needed in a conservation area. It adds that new structures "including fences and walls within the curtilage of a listed building are likely to require planning permission".
The Planning Portal sets the national rule behind every Bridgwater job. It says you will not need to apply for planning permission to erect a new, or alter, maintain, improve or take down an existing fence, wall or gate where it is "next to a highway used by vehicles (or the footpath of such a highway) and it would not exceed one metre in height (from ground level)", or where "it would not exceed two metres in height (from ground level) if elsewhere". Where a Bridgwater boundary already exceeds those limits, the condition is that "its height would not be increased".
Those conditions also require, at a Bridgwater address, that "no part of the site is a listed building or within the curtilage of a listed building" and that "the right to put up or alter fences, walls and gates has not been removed by an article four direction or a planning condition". In a Bridgwater conservation area the Planning Portal says "you might need permission take down a fence, wall or gate".
In Bridgwater we would explain how you check the conservation boundary, any Article 4 direction and the proposed work before confirming whether permission is needed.