The game doesn't punish solo players, but at just level 7 you're not going to have a lot of skill points to make yourself an unstoppable killing machine, and because the difficulty scales with the number of players, even if you played with 3 other level 7 players, you'd all still die at least 7 times.
I started playing with Krieg and I died quite a few times in the low levels, when I had barely any skills, just a pittance of health and low level equipment, but once I advanced through the game and had more skills to invest, I barely ever died except when I press the wrong button and threw a Contraband Sky Rocket grenade at my feet in the indoor shooting range at Sanctuary.
Just perceiver and unlock more skills to give yourself more health, better shields, health recovery or allow you to kill them before they kill you.
It's no that the game is more punishing to solo players, it's just that with the skill progression and character development, low level players with low health, no skills and poor equipment are more vulnerable to the dangers of the game, so when you are a level 7 character who is barely better than a normal human, it's best to act like a normal human and use cover, avoid overwhelming firefights, approach outnumbered battles sensibly, pick off targets one by one from a distance and retreat if necessary to heal.
At level 7, I was a vulnerable grunt who couldn't do much more than take 1% more damage before dying and explode when I died, but now I laugh in the face of danger, actively seek out to get hit and when my health gets low, I just transform into an unstoppable hulking mass of destruction, regain all my health and kill everything in sight... and on the off chance I do get downed, I don't die, but instead just run at enemies throwing dynamite at them so get second winds, or detonate myself in a massive explosion that is sure to kill anything around me and get a second wind... all skills and abilities that I didn't even have a semblance of at level 7.