This is a good opportunity to learn by 'baptism' of fire, and if you can learn fast you can accelerate your path to senior developer. In a bigger org you may be forced to spend years getting to senior developer, but it sounds like you can get there quicker with the responsibility you have here.
There is the stress of dealing with complex code and frameworks as a newbie, with the owner breathing down your neck. So try to negotiate more time for each task, and time for learning general web development. E.g. if they use AngularJS say you need a couple of days to learn it from tutorials etc.
If they are complete assholes about it and pressure you with 'just get it done' 'we need it done by end of day' 'why is this taking so long' then maybe it is time to look elsewhere for a job.
In any case look at this job in terms of if it is good for your career (and more importantly, your mental health!) and if you thrive on high pressure fast learning or prefer to work for a bigger company that has the resources to give you more time.
Out of curiosity how many devs are working there? Is it one man and his dog, and you?