Hi, welcome to the forums...you definitely need to not EVER go back to that vet, there is no way he is a reptile specialist vet if he told you to feed a baby/juvenile dragon live food every other day. That's the total opposite of what you should have been doing, and if he's weak, lethargic, and more importantly he has a bit of MBD from the coil
UVB bulb and then the compact
UVB bulb, maybe a bit stunted but not badly. Hopefully you can get him on a correct live insect schedule, dusting schedule, and get his UVB tube mounted correctly for him so that he starts getting adequate
UVB light.
First off, there should never be a temperature inside his tank anywhere over 110 degrees!!! 115-120 degrees is in the lethal range if he's in it for any length of time, so please move your basking lamp up and away, or possibly even go to a lower wattage bright white basking bulb, and make sure you are using either a Digital Probe Thermometer or a Temperature Gun so that you can actually measure the Basking Spot Surface Temp, you cannot with any type of stick-on thermometer. Basking Spot Surface Temp between 105-110 absolute max (at a year old they tend to like it a bit lower, between 100-105), Hot Side Ambient (air) Temp between 88-93 max, and Cool Side Ambient (air) between 75-80 max.
What Reptisun 10.0 UVB tube did you buy, I can't really see the length, either 18" or a 24" most likely, but is it the T8 or the T5HO strength? If it is the 10.0 T8 strength then it absolutely must be mounted inside his tank and under the mesh lid, as it blocks 40% of the
UVB light, and the T8 is not strong enough to still deliver adequate
UVB light to him. The T8 also must be at least within 6" at a max distance away from his baking spot, and it must be replaced once every 6 months, as that is the age that they stop emitting any
UVB light at all, even though the tube will not be burnt out and will still light up. If you got the 10.0 T5HO strength UVB tube then you can sit it on top of the mesh lid, but it still must be within at least 11" of the basking spot, and it must be replaced once every 10 months to 12 months max. I have been using the 22" Reptisun 10.0 T5HO UVb tubes for years and I still mount them on the inside of the tank at a distance around 9-10" away from their main Basking Spot, and I replace them once every 10 months just to be safe. Also, make sure that both his new UVB tube and his bright white basking light are on every single day for at least 13-14 hours per day, and that you turn ALL lights off at night. It needs to be pitch black in his tank and the temp needs to drop considerably, just like it does in the desert. As long as his tank temp is not below around 65 degrees at night, he needs no nighttime heat source at all (your house would have to be at 60 degrees or below for you to need a nighttime heat source for him, which would be freezing, so most people need no nighttime heat source).
You need to find a place to order your live insects online in bulk, because it is not at all cost effective to buy them at a pet shop, he should be eating a ton of live insects every single day. He needs at least 2 live feeding sessions every single day but 3 is ideal, up until he's 8 months old, then at 8 months you go to 2 live feeding sessions a day. Each session needs to last between 10-15 minutes, where he is allowed to eat as many live insects as he wants to, dropping in 2-3 at a time and only stopping when he walks away to go bask. Crickets, dubia roaches, Phoenix Worms/BSFL, or Silkworms are your acceptable live staple feeders for a dragon, with occasional treat live insects being wax worms, butter worms, or hornworms, maybe 3-4 a week. No mealworms. He should be offered a fresh greens salad every day, but not until after his first live feeding session, and if he eats it it's only a bonus, and should not be considered a part of his daily diet at all until he's a year old.
Dragons continue to grow until they are 2 years old, but they do 80% or more of their growth and development during year one, and they eat basically no vegetation in the wild up until they are over a year old. This is why not many babies or juveniles even touch greens or veggies until over a year old, because they know they don't need it and they don't want it, they need huge amounts of live insects. That's why I suggest you order online in bulk, from a place like
http://www.dubiaroaches.com (they have all live feeders, not just dubia roaches, and they are the cheapest I've seen). For example, my 6 month old male eats large Phoenix Worms/BSFL as his live staple insect, and he eats between 20-30 in the morning, 20-30 in the mid afternoon, and 15-20 in the early evening every single day, 7 days a week. So I buy them buy the 1,000 (I have 3 dragons, one adult, one just turning a year, and the 6 month old). We go through a lot of bugs, but one 4-5 month old beardie should be eating at least around 40 live insects every single day, that are smaller than the space between his eyes. As he grows you can feed him larger sized bugs and then obviously he'll eat less...
I'm sure he's got calcium deficiency going on due to mostly the lack of
UVB light and then due to the poor diet and dusting schedule, so I'd actually be dusting all of his live insects from one of the 2 or 3 daily feeding sessions in calcium 7 days a week for a while, until he's around 8 months old, then go down to all the insects from one of the 2 or 3 daily feeding sessions 5 times a week. He needs the extra calcium under adequate
UVb light to up his bone density and to try to reverse any results of the MBD. The thing he's doing with his leg is most likely due to the MBD, it may not be (they do weird movements and positions sometimes) but based on his history of no adequate
UVB light and poor diet/dusting you need to be on the safe side for now. Dust all the insects from 1 or the 2 or 3 live feeding sessions daily in a multivitamin powder 3 times a week.