The fact that the timing belt is still there does not at all mean that there is no problem with the timing belt; it just means the belt has not broken. But most timing belt failures do not involve the belt breaking; instead; the belt stretches enough that it slips off the proper tooth, and jumps onto a different tooth. This throws the camshaft(s) out of time, and in some engine designs also throws the ignition timing off by the same amount. The cams being out of time will by itself be enough to immediately shut the engine down and prevent restarting. And that event is typically preceded by a "pop."
You can confirm the belt has jumped time by running a compression test. If the compression is less than 120 psi; the belt has probably jumped.
Another test is to remove the distributor cap, and turn the engine until the timing marks line up between TDC and 10 degrees BTDC. With the crankshaft in that position; the distributor rotor should point directly to either the cap terminal for the #1 cylinder plug wire or the #4 cylinder plug wire. If it points anywhere else; the timing belt has jumped.
There are two belts, two tensioners, one idler, and two seals, and there may also be a chain that drives a balance shaft. All of these parts ought to be inspected and replaced if they are worn.
The engine in your car is called an "interference engine." This means that if the belt breaks or jumps out of time; it becomes very likely that one or more valves will strike a piston. If this happens, valves usually becomes bent. In extreme cases; the impact can break or damage a piston. So if the belt jumped; you can expect to have to do far more than just change the belt. The cylinder heads will need to be removed and inspected by a machine shop for damage. The valves will probably need to be reground and or replaced. This work may exceed the value of the car.