I had a small homework on lambda-calculus and Turing machines equivalence when I was an undergrad (during my third year of uni iirc). It's in French but you may be able to follow by looking at the "code" parts. In it I use pure lambda-calculus to simulate a universal Turing Machine. It's probably not done the best not the more canonical way, but it's what I got while discovering lambda-calculus at the same time, so it might be good as a starting point for where you're at now. You can find the paper here: https://pablo.rauzy.name/files/lambdacalcul.pdf. Hope it helps!