IconVault's Turing Machine simulator lets you define states and transition rules (what to read, what to write, which way to move, which state is next) and watch the machine run on a live tape, either step by step or run to completion, with adjustable speed and a reset button.

It ships with working examples, binary increment, palindrome check and flip bits, so you can learn by editing a real machine instead of starting from theory, and everything runs entirely in your browser, free with no signup. The honest note: it is a teaching simulator, not a formal verifier, so it will not prove halting or decidability for you.