The reason for your confusion is that the Rāmāyaṇa translation you are quoting from is adding the phrase thirty-seven times not present in the original Sanskrit verses or the word-to-word translation available on the same website (valmikiramayan.net).
vadham apratirūpam tu pituḥ śhrutvā su dāruṇam |
kṣhatram utsādayan roṣhāt jātam jātam anekaśhaḥ || 1-75-24
pṛthivīm cha akhilām prāpya kāśhyapāya mahātmane |
yajñasya ante tadā rāma dakṣhiṇām puṇya karmaṇe || 1-75-25
dattvā mahendra nilayaḥ tapo bala samanvitaḥ |
24b, c, 25, 26a. raama = oh, Rama; a prati ruupam = not, similar, in form [type, unregenerate]; su daaruNam = highly, gruesome; pituH vadham shrutvaa = father's, murder, on hearing; roSaat = with rancour; jaatam jaatam = newborn, as newborn - as and when born; kSatram = Kshatriya-s; an ekashaH = not, for one time; utsaadayan = extirpating [Kshatriya lineages]; akhilaam pR^ithiviim praapya = in entirety, planet earth, on getting [under my control]; yaGYasya ante = Vedic-ritual, at the end of; mahaatmane = to the divine-souled one; puNya karmaNe = of pious, observances; kaashyapaaya = to Sage Kashyapa; dakSiNaam dattvaa = as ritualistic-generosity, on giving; tadaa = then; tapaH bala samanvitaH = ascesis, powers of, conjoined [with me]; mahendra nilayaH = Mt. Mahendra, indweller [I am at present.]
Oh, Rama, on hearing the unregenerate and highly perfidious murdering of my father, I rancorously extirpated Kshatriya-s as and when they are born, that too not for one time, but I did so for thirty-seven times going around the earth... and on getting the entire earth under my control I performed Vedic-ritual, and at the end of that Vedic-ritual, I gave all that earth to sage Kashyapa, a sage with divine soul and with pious observances, as a ritualistic-generosity... and I am at present on Mt. Mahendra practising ascesis and thus conjoined are the powers of ascesis in me... [1-75-24b, c, 25, 26a]
Here's an alternate translation which doesn't specify the times Paraśurāma had slaughtered all the kṣatriyas.
O Rama! Having heard the slaying of my father incomparable in extreme ruthlessness,
I decimated the Kshatriya race again and again out of fury as they were born and reborn. I conquered the entire earth. At the conclusion of the sacrifice I conferred it on the great soul Kasyapa of meritorious acts. Gifted with ascetic energy I have (now) made the Mahendra mountain my abode.
It is generally believed that Paraśurāma exterminated kṣatriyas 21 times, the Sanskrit phrase being used is triḥ sapta kṛtvaḥ (3 x 7 times, as explained here) so the Bhāgavata Purāṇa translation you quoted is right. The Mahābhārata also uses the same phrase narrating the same incident and K M Ganguli translates it the same way.
Book 1: Adi Parva; Chapter 58
rahasyaṃ khalv idaṃ rājan devānām iti naḥ śrutam
tat tu te kathayiṣyāmi namaskṛtvā svayaṃ bhuve
4 triḥ saptakṛtvaḥ pṛthivīṃ kṛtvā niḥkṣatriyāṃ purā
jāmadagnyas tapas tepe mahendre parvatottame
Vaisampayana said, 'It hath been heard by us, O monarch, that what thou askest is a mystery even to the gods. I shall, however, speak of it unto thee, after bowing down (to the self-born). The son of Jamadagni (Parasurama), after twenty-one times making the earth bereft of Kshatriyas wended to that best of mountains Mahendra and there began his ascetic penances...