I wouldn't judge too much from the visual representation, it was meant to show that its bigger, especially in address space.
In an HTTP or HTTPS connection, which dominate Internet traffic, the vast majority of packets are at the MTU, since the data being transferred far exceeds the MTU. Using IPv4 instead of IPv6 won't make the packets any smaller, it will just leave a few more bytes for data within the MTU envelope, and statistically speaking will sometimes avoid the need for an additional small packet at the end of the transfer.
so, while your system may show you the bits for your visual queues; The binary format as its being modulated to be sent over medium will transform the signal based on bits. If you have lets say source address (i'll use ipv4 as example, same goes on any binary modulation)
10.0.0.1
00001010.00000000.00000000.00000001
The signal sent won't actually reflect this binary code. The signal generator will actually send out this
00001010.0.0.00000001
Some more advanced systems may utilize all kinds of tricks to get it even smaller. This is no longer physically 32bit, but 18bit that CPU will read, and interpret oh yeah this is actually 00001010.00000000.00000000.00000001 or something else... I have simplified that terribly it can get far more advanced, and there are plenty of tricks/hacks that it can be used with - since cpu on switch is quite dumb at the time of reading it.
There were many tricks that were used in past
Few decades ago where you had bigger control over your own router/modem at home from your ISP, changing subnet from /24 to bigger one lets say /16 often completely changed routing your transmission would take, and if network didn't complain / block your traffic would actually be 18 bits vs 25 bits in /24 when being transmitted, you stack few things on top of that and your packets get significantly smaller. Often leading you to networks hops you would normally not get (this in conjunction with BGP protocol - also used in some form of MITM attacks in eBGP attacks can bring down whole net as is - as you could essentially reach networks where you shouldn't be able to reach - but because CPU's are dumb and only calculate binary - one can generate packet that will in fact broadcast advertising route/s to e/BGP edge routers and bring down sites or whole net) Since then its much harder to do any of that, but to some point its still possible to do. (you shouldn't do it or even try -> alpabet friends were visiting people from defcon for years each time someone presented that.)