Unable to log in to some websites on Safari

I am about to start work at two different hospitals. As part of the on-boarding process, I have to sign in to multiple websites like the HR portal and the online learning/ compliance training websites. So far I have had trouble logging in to the HR portal for both hospitals and the online learning portal for each hospital (4 separate websites from separate organizations with different usernames/ passwords). Each website had a different problem that did not seem related to the problems I was having with other websites. One said my user ID was not found. Another linked me to a page it was not supposed to each time I logged in. For each issue, I spent multiple weeks being transferred to various IT people, all of whom were stumped. Finally, I tried signing in to the different websites on firefox and chrome and it worked without any issues for all 4 websites that were giving my problems. I have done rotations at both of these hospitals as a student and therefore had to log into all of these websites in the past on the same computer I am using now. I did not have any problems previously. All of this started happening after I updated my OS. (Still using Mojave but recently updated to 10.14.6). Any ideas? It does not happen on all websites. I am able to log on to some of the websites I need for each hospital but not all of them. Seems to happen more on websites that require a more secure log in with two factor identification but I am still able to log in to some websites that require secure login. Have not had problems with things like facebook, google, ect.