How do diesel submarines not end up killing their crew?

Diesel Electric submarine use electric energy stored in the batteries to make way under water. On the surface they use diesel engines and the engine exhaust is vented outside.

During World War 2 Germans outfitted some of their U-boats with a snorkel that allowed them get fresh air and vent the engine exhaust to the outside while submerged. The U-boats had major disadvantage: they could either charge their batteries or make way in the surface, but not both at the same time. American Submarines used the power train from the FT type diesel electric locomotives where engines drove the generators that produced the electrical power to charge the batteries and drive the electric motors for main propulsion, thus charging batteries while under way.

In the 1990s a number of countries developed Air Independent Propulsion systems for submarines. The French Scorpene-class and its derivatives use a closed steam turbine power plant powered by steam generated by combustion of ethanol and oxygen stored at 80 atmospheres. This pressure allows the exhaust to be expelled outboard at any depth without the use of the exhaust compressor. The Scorpene can stay under water for at least 21 days.

The Swedish Gotland-class boats use a Stirling engine that burns diesel fuel and liquid oxygen. Like Scorpene-class, the Gotland boats can remain submerged for weeks at a time. This system was retrofitted to the Västergötland-class and their Singapore Navy's Archer-class sister boats and is also used on the Japanese Sōryū-class boats.

Another propulsion option is fuel cells using Siemens polymer electrolyte membrane (PEM) hydrogen fuel cells. These are used on the German Navy Type 212, and its export derivative Type 214 and the Dolphin class developed for Israel.

The diesel engines are only used while on the surface, the exhaust gases are released into the air, and fresh required for the engines (and the crew) is drawn in from outside. As well as providing power for the propellers, one or more engines can be used to recharge the banks of batteries.

The batteries are used for driving the submarine while it is submerged. For obvious reasons, the diesel engines are not used.

Some submarines included a snorkel, which permitted the use of diesel engines when the submarine was at periscope depth.

Unlike nuclear submarines, diesel-electric submarines spend the majority of their time on the surface - and only submerge for short periods of time, to attack or avoid a specific threat.

Diesel engines are not run while the sub is under water only when fresh air can be drawn in and exhaust vented to charge the batteries that are used to run the boat submerged. How can you have even looked at the operations of old diesel subs and not learned this?

As it happens, I just went on a tour of the USS Blueback Submarine... they have to surface so they can release the exhaust.