John Gill Commentary Leviticus 5:10

"And he shall offer the second for a burnt-offering, according to the ordinance; and the priest shall make atonement for him as concerning his sin which he hath sinned, and he shall be forgiven." — Leviticus 5:10 (ASV)

And he shall offer the second for a burnt offering,
according to the manner
That is, the second turtledove or young pigeon, after the other was made a sin offering; and the manner according to which this was offered was not according to the rite or manner of the bird chosen first for a sin offering, as the Targum of Jonathan, but according to the burnt offering of the fowl in (Leviticus 1:15–17) so Jarchi and Ben Gersom:

and the priest shall make an atonement for him, for his sin which
he had sinned, and it shall be forgiven him ;
upon the atonement made; and so forgiveness of sin with God proceeds upon the atonement made by the blood of Christ, (Hebrews 9:22) .

God never took one step towards it, without a regard to Christ the propitiation for sin; he promised it with a view to him; there is no instance of pardon under the Old Testament but in this way, and God always has respect to Christ in pardon, it is for his sake; and this way of forgiveness best provides for the glory of the divine perfections; there can be no better way, or infinite wisdom would have used it; there could be no other way, considering the council and covenant of peace; to pardon, without atonement and satisfaction, is not consistent with the purity, justice, and veracity of God; and to observe this great truth, the phrase is afterwards frequently repeated,