The only really known way to tier up is to start playing Ranked rounds (9/18 hole rounds) and getting your AVG down to the necessary numbers. See chart below.
While the chart is correct and this is the most traditional path, fully covered, there are a few others of which we have partial knowledge. Unfortunately, WGT is mute about them.
- Matchplay win against higher tiers
The average is lowered by a certain amount, e.g. -0.7 for a TM winning against a Legend (been there, done that!). Some kind of formula has been published here.
- Credit wins
To win credits in tournaments (?and Matches?) will bring players closer to or over the tier promotion line by a hidden effect on average + saturation no. In such cases, the moment of promotion is not linked to a game's finalisation but to payout time.
- Level promotion on mobile
Mobile players who frequently do not play (many) ranked strokeplay rounds may be promoted regardless of their high average (e.g. 150). It looks like this is connected to certain levels, and it goes up at least to Legend => TL tier promotion at levels beyond 100. See this thread as a starter.