An insurer can’t find anyone at fault
An insurance company cannot find you or anyone else at fault - they don't have the power. They are alleging that you are at fault and, presumably, demanding damages.
Whether you are at fault or not is a matter for you to concede (by paying them) or a court to determine based on the evidence when they sue you (or you sue them for your damages).
The other driver’s insurer only knows what they have been told and it’s likely the other driver genuinely believes they were not at fault and, based on what you say happened, they may very well be right, or at the very least, that both drivers were at fault. For example, it’s not clear if you turned from the left most lane into the left most lane as you are required to do or if you changed lanes during the turn. Notwithstanding, it sounds like you drove into them (that is, the front of your car was behind the front of their car) and the fundamental rule of driving is don’t drive into things - failing to avoid a collision is a go to offense in all driving rules.