The natural idiomatic expression of 「関係を絶つ」 might be 「縁(えん）を切（き）る」.
縁を切る means "break off" "break out with someone" "give someone the elbow".
I wonder "burn your bridges" might have different meaning.
If "burn your bridges" means to make yourself not to going back in order to succeed, to put yourself into the most stressful and dengerous situation in order to make your ability to maximum level,
the perfect translation would be "背水の陣” ”背水（はいすい）の陣（じん）を敷（し）く”.
背水の陣 is the idiom or proverb coming from old China.