Two steps...

Step 1: Get a 121/135 job and start building professional multi-turbine crew time...you're not getting a major airline job without it. To get started you either need 1500 for the regionals, or 400-500 for 135 SIC. Pretty much any type of time will work for that...they're desperate enough to let anyone try the training program these days. Whether you graduate or not will likely come down to how much (recent) IFR/IMC time you have.

Step 2: Build 2000+ turbine with ideally 1000+ TPIC in 121/135. LCC's will consider non-121 pilots but frankly you're probably at a disadvantage compared to regional pilots. If for no other reason than regional pilots all have recommendations from their CA's who have moved on to majors.

Time in smaller turboprops isn't going to weigh as heavily with the majors as glass/jet time. Some majors are OK with it, you'll just have to figure out who they are.