Can the floor framing in my 1930s bathroom support my tub and shower?
If I assume you have 2x12 joists, then it's all good. If you want it better you can add full depth joist blocking perpendicular to the joists to try and help the floor out a little, but you don't have to.

If I assume you have 2x10 joists, then it's all good except for that joist under the shower/tub. It probably wouldn't catastrophically fail, but add a second joist next to it if possible. It'll also help prevent your new tile job from cracking by decreasing the live load deflection you'll get when filling the tub. If you absolutely can't add another joist there, you can add full depth joist blocking perpendicular to the joists to try and help the floor out a little.

If I assume you have 2x8 joists, that's bad. I hope you don't have 2x8 joists!