Is that 14 gauge solid wire (the kind you put in the walls of your house) or normal (flexible) wire? Low voltage electricity doesn't work well with solid-core wire; it's the same effect as running a noticeably smaller gauge wire. I wouldn't have even thought you might be using household wire if you hadn't mentioned using a light switch as your power switch. For similar reasons, I would think the switch isn't going to be the most efficient, either.
In theory, you're pulling about 15 amps (350w / 24v = 14.583a). Depending on how long the wire is, I'd recommend 12 or better yet, 10 gauge automotive-type(or something similar) wire. The longer the wire, the bigger it should be, so if your switch is at your handlebars and your motor and batteries are at the back of the bike, go for the 10 gauge. Better yet, use an automotive relay, so the wire between the batteries and the motor is only inches long, and use a remote switch to turn the relay on and off. That would be the most efficient way of doing it. You could use a tiny/discreet wire running to the switch then, if aesthetics matter.
There are reasons behind my recommendations(no, really, there is!). Efficiency being one of them. Drawing more current than a given wire can comfortable handle will make the wire hot, and said heat is wasted electricity. The other being that your motor won't get the power it's demanding (especially under load) - the voltage at the motor will be less and therefore so will the amps. Your motor now has to work harder than it should, running less efficiently and wasting more power as heat. More pwoer as heat makes your motor burn out much sooner.
Well, that's the first half of my lecture on electric basics done, now to throw chemistry into the mix. For all I know, I'm wasting my breath and just annoying you and everyone else, so feel free to tell me to shut up if I'm not providing any useful information.
So...batteries. You're running 2x7.5amp-hour, 12-volt batteries. And pulling up to 15 amps out of them. So, in theory, at full draw, they should last half an hour...but they won't last half an hour under that load. They'll last 15 minutes, most likely. The faster you draw said amps out of the battery (lead-acid/SLA at least), the fewer amp-hours you actually get. In other words, the greater the draw, the less efficient the batteries. It's also hard on the batteries, draining them too fast, and they will have a much shorter life span.
In other words, I think the batteries are rather undersized for the job. Not so much because you need twice as many amp-hours but more because you're losing half of said amp-hours to the inefficiency of the batteries. Larger batteries can put out more amps with less efficiency loss. If you switch to 12 amp-hour batteries, you won't get 1.6 times as much range, you'll probably get 2 times or even more. The next size up from that would be optimal, they come in 15/18/20 amp-hour, depending on the manufacturer/quality. And rather than 2 to 2.7 times as far as your current batteries take you, you'd be looking at more like 3 to 4 times the range. And the batteries would have a much longer life span too. Or a second bank of 7.5ah batteries in parallel would have the same effect. (aiming for the batteries to last 1 to 2 hours under full load is an easy way to decide what size batteries to go for)
Sheesh. That was a long way of pointing out that your wiring might be inefficient, and that your batteries are (almost certainly) undersized for the job. I'll shut up now.