The r arrived later, reshaping an older word for man.
Old English brȳdguma became Middle English bridegome. Merriam-Webster traces the Scots form brydegrome to folk etymology: an r entered the second part. Bridegroom preserves that reshaped ending.
These are selected forms in a derivation, not exclusive spelling periods.
Source
Merriam-Webster: bridegroomWord History gives the sequence, the Scots form, folk-etymological reshaping and guma meaning man.