I see myself in her overcoming her difficult past and moving forward.
I identify with her in how she overcomes her difficult past and moves on with her life.
つらい過去 = difficult past
乗り越える = overcome
次に進む = move forward / move on with one’s life
自分を重ねる = I see myself in / I identify with
“identify” は「突き止める」と「見分ける」の意味もあります。例えば “They identified the criminal using security camera footage.” (防犯カメラの映像で犯人を突き止めた。) 似ている英語の “identity” (アイデンティティ)もあります。
I saw myself in her as she moved on from her painful past.
・I saw myself in her as she moved on from her painful past.
「つらい過去を乗り越え次に進む彼女の姿に自分を重ねた」
see myself in her(彼女に自分を重ねた)と move on from her painful past(つらい過去から次に進む)を組み合わせることで表現できます。
・I related to her as she overcame her difficult past and took a step forward.
「困難な過去を乗り越えて一歩を踏み出す(次に進む)彼女の姿に自分を重ねた」という意味の表現です。