Loyalty pressure is generated largely by population. Invading another continent requires having multiple cities with decent-sized populations reinforcing each other's loyalty. If you conquer multiple cities close together simultaneously or in quick succession, it should be easier to hold them.
Depending on the map circumstances, you can also settle cities on nearby islands or on parts of the continent that the other civs haven't settled yet, in preparation for invading those civs. These cities will then provide a loyalty buffer for your conquests further into the continent. Again, you'll want multiple cities settled simultaneously or in quick succession, and you'll want to get their populations up as quickly as possible, in order to prevent them from going independent.
Use policies such as Colonial Offices to boost your loyalty and populations quickly. Use domestic trade routes with large food incomes to quickly raise the populations of the cities. Slot a governor and a military unit in the city and have Limitanei and Praetorium policies (or later-game equivalent) to further buff loyalty. You should also save up a large sum of gold prior to conquering or settling the new cities on the new continent. Spend that money to buy a Monument and Granary in each city. Put Reyna governor with the Contractor promotion in one of the cities and buy an Entertainment complex or Water Park as soon as you can, then run Bread and Circuses.
Being in a Golden Age definitely helps. Having the Hic Sunt Draconis dedication bonus will start newly-settled cities on other continents with extra population and loyalty. You probably shouldn't bother trying to invade another continent if you're in a dark age, as the loyalty penalty will probably flip your cities within just a few turns.
If you're able to quickly capture a few additional cities, you can raze cities along the border with your newly-conquered cities in order to reduce the loyalty pressure that the opponent is exerting on your conquered cities.
I've never tried this, so I don't know for sure if it will work, but if you don't want to raze the cities (because they have valuable wonders, districts, or whatever) you can maybe try gifting them to a friendly or neutral third-party civ. If you gift them to a civ that is in a Dark Age, they might go independent within just a few turns, at which point they'll stop exerting loyalty against your cities, and you can re-capture them on a later turn with no additional grievances. Capturing a single city multiple times, however, will reduce its population each time it's captured, lowering how much loyalty it will give you, and (obviously) reducing its productivity, when you finally do get around to annexing it.