Yup, it is normal to be able to see the case bulge where the bullet is seated.
As long as your loaded rounds outside case diameter measures less than .379" (ref. SAAMI drawing specs for the .38Spcl & .357Mag) and they fit in the chamber okay then you are fine.
When you're loading lead, you should still see that bulge too since most lead bullets for the .38/.357 are .358 diameter where jacket bullets are spec'd to .357" diameter. When seating the softer lead bullets you might displace a bit of the lead and swage down the bullet down to a little under the .358" diameter.
You might be interested in trying some of the copper plated bullets instead of jacketed. They're a little more expensive than plain lead but cheaper than jacketed, and they pretty much eliminate the leading problem.
for one example, but there are other brands of plated bullets too.