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I slipped through the crowd by turning my body sideways.
Literal
Crowd [path-を] body [object-を] sideways-[into-に] do slipped-through.
Two を particles in sequence with different roles — 人込みを marks the crowd as a space being traversed (path-of-motion を, like 道を歩く 'walk along a road'), and 体を marks the body as the direct object of 横にする 'make sideways.' 横にする uses the productive X + に + する pattern 'to make X into Y.' The te-form subordinate clause 体を横にして describes the manner of traversing the crowd — 'by turning the body sideways.'