All I did was change the rear tyre and now the rear derailleur can only access the highest 4 sprockets out of 7. I have followed Parktool videos on adjustments (including pulling the cable probably tighter than I should) to no avail. Adjusting the L Limit Screw at the lowest gear doesn't show any movement to the derailleur. The great change lever won't go any further to low gears either and shows such high resistance when trying that I'm afraid of snapping the cable. What else can I try to fix this?
So - the wheel was removed and reinstalled, the gearing didn't line up, so you started changing things.
Instead, it would have been better to try resolving the indexing problem straight off rather than changing things. Now you've got extra problems to resolve. I would have suggested
- Confirm visually that all the gear wire outer segments are firmly seated in the frame-stops. They can pop out when tension drops, same goes for brake wires.
- Remove/reinstall the wheel and see if that helps. There may be a washer that has swapped sides of the dropout.
All this is a lot easier if you can hang or suspend your bike with the rear wheel off the ground. Rope or anything can help here.
Limit screws do exactly one thing. They limit how far the derailleur can move and have no effect on indexing or gear/chain alignment.
H is Hi and confusingly means the littlest cog, and L for Low means the biggest cog.
Set them up first and then don't touch them again. Don't worry if the chain is a bit clattery at this point, it should move to, and not past, the respective cog. You can manually pull any exposed gear wire to actuate shifting, while slow-pedalling with your right hand on the right pedal. The Low limit screw should be only just allowing the chain onto the largest cog - otherwise the chain can jump over the top and make a mess of spokes. The Hi limit screw is more forgiving.
Then set the barrel adjuster to about half-way in/out. Approx is fine.
Set your right-hand shifter into the "second hardest" gear, meaning not the outside smallest, but the one beside it. The chain should be on the second smallest cog too.
Without pedalling, click your shifter into the hardest gear but don't move the chain
Now loosen the pinch bolt/nut on the rear derailleur, pull through as much of the inner wire as you reasonably can and reclamp the wire.
This should get you into the right ballpark area.
Now the only place to fine-adjust is the rear derailleur's barrel adjuster.
When viewed from the rear (like your photo above) then unscrewing it will lengthen the outer housing and making the derailleur shift toward the middle of the bike, or toward the easier gears.
Likewise, tightening the barrel adjuster will move the derailleur outboard, toward the smaller cogs/harder gears.
In your photo above, you'd want to tighten the barrel adjuster a turn or so, but this presumes the limit screws haven't been fiddled too hard.
You can turn the barrel adjuster with the left hand as you slowly hand-pedal with the right hand on the right pedal. Do mind out for pinch points - there's a lot going on down that area of the bike and you can do a lot of damage to fingers and any dangling things like cuffs and hair.
When you have it perfect, run up and down the cassette using the shifter several times. There's always some compromise. Do be aware that shifting is subtly different when freewheeling vs when there's a load on the bike.
@Criggie has the correct protocol for setting up and indexing a rear derailleur. I'll just add a few things. You need to start over from the beginning so loosen the cable pinch bolt and completely disconnect the inner cable. This allows you to set the H limit without cable tension possibly holding the derailleur from it's full travel to the right. First, though, make sure that your rear wheel is fully within and centered in the drop outs. The wheel and thus the cassette sprocket need to be in correct alignment for indexing to work properly.
You have barrel adjusters--one where the cable enters the rear derailleur and the other where the cable exits the shifter. Begin with each of these adjusters about 2 full turns out from fully in. This will give you room to adjust cable tension both ways (increase cable tension by counter-clockwise rotation, decrease cable tension by clockwise rotation. You'll note that these adjustments will also move the derailleur's guide pulley in or out when the cable is reattached, allowing you to align the guide pulley with the correct cassette cog. In your photo, there is a great deal of misalignment as noted by the sharp bends in the chain. A properly indexed derailleur running the chain on the correct cog will yield a straight chain between the derailleur's and from the cassette cog to the guide pulley.
In your photo, it appears the limit screws are nearly fully wound in and it's likely your derailleur is too limited and not getting far enough in either direction. Check the wheel, disconnect the cable and begin by setting the high limit screw: proper adjustment will be when the guide pulley is right under the outside plane of the small cog.