As always, #Your Headcanon May Vary, these are just my own opinions, and I am SURE other ppl have other excellent answers for this. 

I’m focusing on Louis’s strength at turning and the first few years after, since he does grow more powerful over time on his own (and he also gets *~upgraded~* later in canon, idk whether you accept later canon but it happens!).

TL;DR: I don’t think Lestat purposely made Louis weak, it was a combination of factors, but mostly that the procedure isn’t an exact science*, Lestat was a young maker and turned Louis too soon after making two fledglings before, and the fact that Louis was malnourished** (refusal to kill ppl) for those first few years might have been a contributing factor.

**So Lestat does talk about Louis being weak in IWTV, but not that he purposely made him that way, just that he allowed it to go uncorrected; he didn’t force Louis to kill ppl, or force Louis to embrace his vampiric gifts and learn how to use them:

“Lestat looked at me. ‘I expected you to feel these things
instinctually, as I did,’ he said. When I gave you that first kill, I
thought you would hunger for the next and the next, that you would
go to each human life as if to a full cup, the way I had. But you didn’t.
And all this time I suppose I kept from straightening you out because
you were best weaker. I’d watch you playing shadow in the night,
staring at the falling rain, and I’d think, He’s easy to manage, he’s
simple. But you’re weak, Louis. You’re a mark. For vampires and
now for humans alike. This thing with Babette has exposed us both.
It’s as if you want us both to be destroyed.‘”

^Lestat is saying Louis is weak by choice, and he’s describing weakness of character more than physical strength, so I believe he was physically weak bc of the Dark Gift. The Dark Gift is not an exact science*, despite all good/bad intentions, it’s the vampiric form of pregnancy. But there are things about the procedure that can affect the strength of the fledgling: 

• Whether the blood is transferred once (for Louis) or multiple times (for Marius) between maker and fledgling – Multiple times seems to make a stronger fledgling. Why didn’t Lestat do it multiple times for Louis? I think Louis was already in such bad physical shape that Lestat didn’t want to risk it. Plus, he had already turned 2 fledglings using the single transfer procedure, he probably felt like that was good enough.
• Age, power, and timing of making previous fledglings of the maker –  it seems like power is outweighed by the other two factors.
• The fledgling’s diet after turning – Louis was feeding on animals for the first 4-ish years of vampiring, which is like bad junk food, and probably not drinking

  the volume of blood he needed, either. 

Yes, post-QOTD (and pre-Merrick), Louis refused to drink Lestat’s blood. I headcanon that that felt like a rejection of Lestat bc blood-sharing is a major expression of intimacy for vampires. He might have refused it bc he saw how it had changed Lestat and he didn’t want that to happen to himself, but I think he also wanted to preserve his own vulnerability, in case he wanted to suicide ;A; 

Hit the jump for more, cut for length.

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“So I’m confused, was Louis weak because Lestat made him that way on purpose?” 

I don’t think it was on purpose. It’s not an exact science* and Lestat had only done it 2x, had only heard about the procedure from Armand and Marius. There is some speculation that the blood transfer needs to be exchanged more than once to ensure a stronger fledgling. Marius, for example, exchanged blood with his maker multiple times when he was turned, but with Louis, Lestat only did it once. I think that’s because Louis was so weakened by the bloodletting he’d been forced to undergo (”When
I was subdued finally, and exhausted then almost to the point of death,
they bled me. The fools.”) that Lestat didn’t want to risk exchanging more than once? Idk.

Also, Louis was feeding on animals for the first 4-ish years of vampiring, and that’s like bad junk food. He was probably not even drinking the volume of blood he needed, either. That might have had an impact on his strength. It probably contributed to his attitude at the time, being underfed and undernourished for so long ;A;

“Or was he weak because he was Lestat’s third fledgling in a decade?” 

This is probably more of the reason. As Marius tells Lestat in TVL:

“Well, for one thing, ” he said, “your powers are extraordinary, but
you can’t expect those you make in the next fifty years to equal you or
Gabrielle. Your second child didn’t have half Gabrielle’s strength and
later children will have even less. The blood I gave you will make some difference. If you drink… if you drink from Akasha and Enkil,
which you may choose not to do… that will make some difference
too. But no matter, only so many children can be made by one in a
century. And new offspring will be weak. However, this is not
necessarily a bad thing. The rule of the old covens had wisdom in it
that strength should come with time. And then again, there is the old
truth: you might make titans or imbeciles, no one knows why or how.

In Ricean vampire physiology, a maker needs to wait a good long while between making fledglings; too much frequency will make subsequent fledglings weaker than they could have been. Plus, even though Lestat had the blood of a much older and stronger vampire when he was turned (Magnus), Lestat himself was only a decade into vampiring himself. It seems the vampiric spirit discourages the transfer of powers from young vampires to their fledglings. If anyone got the bulk of that power, it was Gabrielle, Lestat’s first.

“Wouldn’t he be stronger because Lestat had Akasha’s blood in him?”

Marius said that that would make “some difference” but I think the fact that Lestat had already turned 2 vampires, and was young still himself, prevented that power from being transferred.

“I know Louis refused to drink from Lestat, was he ultimately weak because he chose to be?” 

He was weak by vampire standards at first, and yes, I think he chose to remain that way. But he’s still stronger and faster than a mortal. He’s also able to defend himself and kick a lot of ass. What he lacks physically he makes up for mentally, he’s strategic in the way he attacks when he does attack, and he can hold his own against much stronger and older vampires (he took out most of the Theatre des Vampires on his own in IWTV!).

*Re: the Dark Trick is not an exact science: 

Armand mentions in TVL:

But let Armand understand here also that the effect of the Dark Trick is unpredictable, even when passed on by the very young vampire and with all due care. For reasons no one knows, some mortals when Born to Darkness become as powerful as Titans, others may be no more than corpses that move. That is why mortals must be chosen with skill. Those with great passion and indomitable will should be avoided as well as those who have none.