DISCLAIMER: Due to the nature of your question, the fact that is your very first post on the forum, as well as a high volume of new firearms owners emerging in the wake of the Corona Virus (COVID-19) outbreak, I am assuming that you are inexperienced with firearms or at least with revolvers, ergo the following is written under the aforementioned assumption. Reader discretion is advised.
No, it shouldn't be able to chamber, and even if it somehow can, it still shouldn't be done.
.38 Super Auto is a semiautomatic pistol cartridge which isn't related to .38 Special nor .357 Magnum, it's basically a turbocharged version of the obsolete .38 Automatic Colt Pistol cartridge. (Not to be confused with .380 ACP, which is entirely different cartridge, which also cannot/should not be fired from a Taurus 605 or any other .38 Special/.357 Magnum Revolver.)
Generally speaking, you should never attempt to fire any ammunition out of a firearm that isn't chambered for it or otherwise isn't rated for it. Plenty of cartridges are similar enough in dimension to chamber in firearms they were intended to be fired from, but it's typically a recipe for disaster, one way or another, so just don't do it.
Get some proper .38 Special or .357 Magnum because that's what the gun was chambered in. (No other cartridges with ".38" or ".357" in their name can or otherwise ought to be fired from your Taurus 605.)
In closing, there are lots of cartridges out there with similar names, but very few of them are interchangeable, including ones which are just shorter versions of existing cartridges, but you're ONLY supposed to use cartridges which are intended to be shot from the firearm. (Information regarding which cartridges can be fired from your firearm are generally available online, in the owners manual, and should be engraved somewhere on the firearm's barrel.)