She was wearing such thin clothes that catching a cold would have been no surprise.

Literal

She [topic-は] cold [object-を] catch only-natural [quotative-という] thin clothes [object-を] was-wearing.

風邪を引いて当然というような — 'the kind that would make catching a cold only natural.' This complex relative clause modifies 薄い服, showing deep embedding. ~て当然 means 'it's only natural that,' and という wraps it into a modifier.