This is known as clamping. Clamping only occurs for a very small margin of players whose MMR(ELO) is far lower than the expected MMR of the promotion Tier. This is done to ensure that players who advanced to the next tier are properly qualified for a permanent ranking change.
To compare it back to the old system, that would be like a 700 ELO player trying to get into Silver. Even with an awesome streak, the player would have still only gained about 200 ELO, which puts them at 900 ELO and still below Bronze. If they somehow have an amazing ungodly streak and make it to 1400 ELO for Silver, they can still drop back down.
With the new system, instead of going from 700 to 900, it gives the chance to earn enough LP to get into a promotion series, even if your MMR is lower than expected. In this case though, because you're still a bit lower than the Tier, it does make it much more challenging. The benefit of this though, is you can never drop back down.
Admittedly it does feel a little bit more like a letdown in the new system because you can't see that you've raised your ELO, all you see is that you missed Silver. But in reality, you're still better off because you've been given the opportunity to shoot for silver even though you were no where near what you would have needed for Season 1 or Season 2.
If it's encouraging at all, Yegg and his team are working on tweaking the system so clamping won't feel quite as harsh. This will mostly be done through adjusting your early LP gains and losses so they're not as drastic when you get closer to 100 LP.
The other plus is, winning all those games was still a major boost for you. Those wins did significantly push up your MMR which will help lessen or eventually remove the clamping as you continue to move forward.